Properties

Label 8-132e4-1.1-c11e4-0-2
Degree $8$
Conductor $303595776$
Sign $1$
Analytic cond. $1.05807\times 10^{8}$
Root an. cond. $10.0708$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 972·3-s − 7.02e3·5-s − 1.09e4·7-s + 5.90e5·9-s + 6.44e5·11-s − 2.55e6·13-s − 6.82e6·15-s − 1.60e6·17-s − 5.62e6·19-s − 1.06e7·21-s + 1.90e7·23-s − 6.57e7·25-s + 2.86e8·27-s + 1.46e8·29-s − 1.58e8·31-s + 6.26e8·33-s + 7.67e7·35-s − 7.70e8·37-s − 2.48e9·39-s − 5.82e8·41-s − 2.20e9·43-s − 4.14e9·45-s − 1.69e9·47-s − 4.56e9·49-s − 1.56e9·51-s − 2.34e9·53-s − 4.52e9·55-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.00·5-s − 0.245·7-s + 10/3·9-s + 1.20·11-s − 1.91·13-s − 2.32·15-s − 0.274·17-s − 0.520·19-s − 0.567·21-s + 0.616·23-s − 1.34·25-s + 3.84·27-s + 1.32·29-s − 0.991·31-s + 2.78·33-s + 0.247·35-s − 1.82·37-s − 4.41·39-s − 0.785·41-s − 2.28·43-s − 3.34·45-s − 1.07·47-s − 2.31·49-s − 0.633·51-s − 0.769·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1.05807\times 10^{8}\)
Root analytic conductor: \(10.0708\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 11^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p^{5} T )^{4} \)
11$C_1$ \( ( 1 - p^{5} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 1404 p T + 4600144 p^{2} T^{2} + 5901350484 p^{3} T^{3} + 13265126024558 p^{4} T^{4} + 5901350484 p^{14} T^{5} + 4600144 p^{24} T^{6} + 1404 p^{34} T^{7} + p^{44} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 1562 p T + 4688581336 T^{2} + 5691566551066 p T^{3} + 36509883409776386 p^{3} T^{4} + 5691566551066 p^{12} T^{5} + 4688581336 p^{22} T^{6} + 1562 p^{34} T^{7} + p^{44} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2559568 T + 8404664496504 T^{2} + 13270962856518213440 T^{3} + \)\(23\!\cdots\!86\)\( T^{4} + 13270962856518213440 p^{11} T^{5} + 8404664496504 p^{22} T^{6} + 2559568 p^{33} T^{7} + p^{44} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 1606814 T + 7630334892696 T^{2} - \)\(24\!\cdots\!34\)\( T^{3} - \)\(19\!\cdots\!42\)\( T^{4} - \)\(24\!\cdots\!34\)\( p^{11} T^{5} + 7630334892696 p^{22} T^{6} + 1606814 p^{33} T^{7} + p^{44} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 5621902 T + 201491177486628 T^{2} + \)\(15\!\cdots\!90\)\( T^{3} + \)\(35\!\cdots\!86\)\( T^{4} + \)\(15\!\cdots\!90\)\( p^{11} T^{5} + 201491177486628 p^{22} T^{6} + 5621902 p^{33} T^{7} + p^{44} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 19030288 T + 1663250597588888 T^{2} - \)\(68\!\cdots\!28\)\( T^{3} + \)\(15\!\cdots\!94\)\( T^{4} - \)\(68\!\cdots\!28\)\( p^{11} T^{5} + 1663250597588888 p^{22} T^{6} - 19030288 p^{33} T^{7} + p^{44} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 146818034 T + 48968237102861888 T^{2} - \)\(49\!\cdots\!30\)\( T^{3} + \)\(88\!\cdots\!02\)\( T^{4} - \)\(49\!\cdots\!30\)\( p^{11} T^{5} + 48968237102861888 p^{22} T^{6} - 146818034 p^{33} T^{7} + p^{44} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 158103664 T + 94978863162874428 T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(35\!\cdots\!42\)\( T^{4} + \)\(11\!\cdots\!16\)\( p^{11} T^{5} + 94978863162874428 p^{22} T^{6} + 158103664 p^{33} T^{7} + p^{44} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 770947012 T + 760610373721319028 T^{2} + \)\(38\!\cdots\!08\)\( T^{3} + \)\(20\!\cdots\!90\)\( T^{4} + \)\(38\!\cdots\!08\)\( p^{11} T^{5} + 760610373721319028 p^{22} T^{6} + 770947012 p^{33} T^{7} + p^{44} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 582529706 T + 1448248269812918816 T^{2} + \)\(10\!\cdots\!06\)\( T^{3} + \)\(98\!\cdots\!10\)\( T^{4} + \)\(10\!\cdots\!06\)\( p^{11} T^{5} + 1448248269812918816 p^{22} T^{6} + 582529706 p^{33} T^{7} + p^{44} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 2203144746 T + 4731283973047149332 T^{2} + \)\(62\!\cdots\!50\)\( T^{3} + \)\(70\!\cdots\!50\)\( T^{4} + \)\(62\!\cdots\!50\)\( p^{11} T^{5} + 4731283973047149332 p^{22} T^{6} + 2203144746 p^{33} T^{7} + p^{44} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 1693221664 T + 4628248597643526392 T^{2} + \)\(59\!\cdots\!48\)\( T^{3} + \)\(15\!\cdots\!14\)\( T^{4} + \)\(59\!\cdots\!48\)\( p^{11} T^{5} + 4628248597643526392 p^{22} T^{6} + 1693221664 p^{33} T^{7} + p^{44} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 2342022320 T + 7841079369721509192 T^{2} + \)\(46\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!22\)\( T^{4} + \)\(46\!\cdots\!72\)\( p^{11} T^{5} + 7841079369721509192 p^{22} T^{6} + 2342022320 p^{33} T^{7} + p^{44} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 147720480 p T + 98346665480305825628 T^{2} + \)\(48\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!58\)\( T^{4} + \)\(48\!\cdots\!60\)\( p^{11} T^{5} + 98346665480305825628 p^{22} T^{6} + 147720480 p^{34} T^{7} + p^{44} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 2069677236 T + \)\(15\!\cdots\!68\)\( T^{2} + \)\(27\!\cdots\!88\)\( T^{3} + \)\(93\!\cdots\!54\)\( T^{4} + \)\(27\!\cdots\!88\)\( p^{11} T^{5} + \)\(15\!\cdots\!68\)\( p^{22} T^{6} + 2069677236 p^{33} T^{7} + p^{44} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 5402316304 T + \)\(30\!\cdots\!24\)\( T^{2} + \)\(67\!\cdots\!72\)\( T^{3} + \)\(43\!\cdots\!18\)\( T^{4} + \)\(67\!\cdots\!72\)\( p^{11} T^{5} + \)\(30\!\cdots\!24\)\( p^{22} T^{6} + 5402316304 p^{33} T^{7} + p^{44} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 13967424104 T + \)\(57\!\cdots\!08\)\( T^{2} + \)\(33\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!54\)\( T^{4} + \)\(33\!\cdots\!92\)\( p^{11} T^{5} + \)\(57\!\cdots\!08\)\( p^{22} T^{6} + 13967424104 p^{33} T^{7} + p^{44} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 3355935748 T + \)\(52\!\cdots\!80\)\( T^{2} + \)\(20\!\cdots\!08\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} + \)\(20\!\cdots\!08\)\( p^{11} T^{5} + \)\(52\!\cdots\!80\)\( p^{22} T^{6} - 3355935748 p^{33} T^{7} + p^{44} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 32830867762 T + \)\(38\!\cdots\!12\)\( T^{2} - \)\(23\!\cdots\!54\)\( p T^{3} + \)\(75\!\cdots\!54\)\( T^{4} - \)\(23\!\cdots\!54\)\( p^{12} T^{5} + \)\(38\!\cdots\!12\)\( p^{22} T^{6} - 32830867762 p^{33} T^{7} + p^{44} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 10761308844 T + \)\(19\!\cdots\!96\)\( T^{2} + \)\(14\!\cdots\!88\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!88\)\( p^{11} T^{5} + \)\(19\!\cdots\!96\)\( p^{22} T^{6} - 10761308844 p^{33} T^{7} + p^{44} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 51453668352 T + \)\(70\!\cdots\!56\)\( T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(23\!\cdots\!50\)\( T^{4} + \)\(21\!\cdots\!32\)\( p^{11} T^{5} + \)\(70\!\cdots\!56\)\( p^{22} T^{6} + 51453668352 p^{33} T^{7} + p^{44} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 118256509380 T + \)\(97\!\cdots\!44\)\( T^{2} + \)\(23\!\cdots\!32\)\( T^{3} + \)\(30\!\cdots\!46\)\( T^{4} + \)\(23\!\cdots\!32\)\( p^{11} T^{5} + \)\(97\!\cdots\!44\)\( p^{22} T^{6} + 118256509380 p^{33} T^{7} + p^{44} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.389555648531741976654053781465, −7.68597994782226181046157097167, −7.59746312816066058385142484907, −7.56008783060204073585390611028, −7.39031601552053982133927174143, −6.62452655117768962445899546553, −6.57143793445040790383589217393, −6.40669880590385707464111558601, −6.38863598438517771750216723441, −5.27870343786287332705280540468, −5.01329909182053349955686310469, −5.01277642067041840399424814152, −4.79272244520031874439202627142, −4.07205877421680428199847474281, −3.95723620481747265276664017506, −3.70360158024752499977001716801, −3.66197119365045396724221564147, −3.01777125817517305723143215915, −2.85766336448059082951798440977, −2.70560735094918221187569840282, −2.29799230484950356161337144219, −1.80122204702412542438164308805, −1.54979149278471212114022094917, −1.32427978683972853855645699077, −1.30266198794568331057402056175, 0, 0, 0, 0, 1.30266198794568331057402056175, 1.32427978683972853855645699077, 1.54979149278471212114022094917, 1.80122204702412542438164308805, 2.29799230484950356161337144219, 2.70560735094918221187569840282, 2.85766336448059082951798440977, 3.01777125817517305723143215915, 3.66197119365045396724221564147, 3.70360158024752499977001716801, 3.95723620481747265276664017506, 4.07205877421680428199847474281, 4.79272244520031874439202627142, 5.01277642067041840399424814152, 5.01329909182053349955686310469, 5.27870343786287332705280540468, 6.38863598438517771750216723441, 6.40669880590385707464111558601, 6.57143793445040790383589217393, 6.62452655117768962445899546553, 7.39031601552053982133927174143, 7.56008783060204073585390611028, 7.59746312816066058385142484907, 7.68597994782226181046157097167, 8.389555648531741976654053781465

Graph of the $Z$-function along the critical line