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Properties

Label 8-624e4-1.1-c5e4-0-5
Degree $8$
Conductor $151613669376$
Sign $1$
Analytic cond. $1.00318\times 10^{8}$
Root an. cond. $10.0039$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 36·3-s − 8·5-s + 176·7-s + 810·9-s − 96·11-s − 676·13-s + 288·15-s + 1.22e3·17-s − 3.85e3·19-s − 6.33e3·21-s + 368·23-s − 3.27e3·25-s − 1.45e4·27-s + 264·29-s − 1.09e4·31-s + 3.45e3·33-s − 1.40e3·35-s + 1.20e4·37-s + 2.43e4·39-s + 3.27e4·41-s − 2.42e4·43-s − 6.48e3·45-s − 1.41e4·47-s + 2.34e3·49-s − 4.40e4·51-s + 3.73e4·53-s + 768·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.143·5-s + 1.35·7-s + 10/3·9-s − 0.239·11-s − 1.10·13-s + 0.330·15-s + 1.02·17-s − 2.45·19-s − 3.13·21-s + 0.145·23-s − 1.04·25-s − 3.84·27-s + 0.0582·29-s − 2.04·31-s + 0.552·33-s − 0.194·35-s + 1.44·37-s + 2.56·39-s + 3.04·41-s − 2.00·43-s − 0.477·45-s − 0.936·47-s + 0.139·49-s − 2.37·51-s + 1.82·53-s + 0.0342·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.00318\times 10^{8}\)
Root analytic conductor: \(10.0039\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 13^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
13$C_1$ \( ( 1 + p^{2} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 8 T + 668 p T^{2} - 16808 T^{3} + 5971334 T^{4} - 16808 p^{5} T^{5} + 668 p^{11} T^{6} + 8 p^{15} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 176 T + 28628 T^{2} - 3721296 T^{3} + 605143094 T^{4} - 3721296 p^{5} T^{5} + 28628 p^{10} T^{6} - 176 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 96 T + 472148 T^{2} + 3836512 p T^{3} + 101801657814 T^{4} + 3836512 p^{6} T^{5} + 472148 p^{10} T^{6} + 96 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 72 p T + 4103356 T^{2} - 4084243320 T^{3} + 8525013382982 T^{4} - 4084243320 p^{5} T^{5} + 4103356 p^{10} T^{6} - 72 p^{16} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 3856 T + 10968900 T^{2} + 21938174576 T^{3} + 37174174246918 T^{4} + 21938174576 p^{5} T^{5} + 10968900 p^{10} T^{6} + 3856 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 16 p T + 7998332 T^{2} + 18196605392 T^{3} + 21476708847014 T^{4} + 18196605392 p^{5} T^{5} + 7998332 p^{10} T^{6} - 16 p^{16} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 264 T + 525052 p T^{2} + 15901448616 T^{3} + 746777890351734 T^{4} + 15901448616 p^{5} T^{5} + 525052 p^{11} T^{6} - 264 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 10928 T + 105783700 T^{2} + 19405988912 p T^{3} + 3743626337366998 T^{4} + 19405988912 p^{6} T^{5} + 105783700 p^{10} T^{6} + 10928 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 12008 T + 101704620 T^{2} - 557784338872 T^{3} + 5983242191546326 T^{4} - 557784338872 p^{5} T^{5} + 101704620 p^{10} T^{6} - 12008 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 32792 T + 727984380 T^{2} - 11248602324648 T^{3} + 139994437031392918 T^{4} - 11248602324648 p^{5} T^{5} + 727984380 p^{10} T^{6} - 32792 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 24288 T + 790918124 T^{2} + 11423099410656 T^{3} + 190216374004423830 T^{4} + 11423099410656 p^{5} T^{5} + 790918124 p^{10} T^{6} + 24288 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 14176 T + 735750212 T^{2} + 8384297776032 T^{3} + 239936046865012998 T^{4} + 8384297776032 p^{5} T^{5} + 735750212 p^{10} T^{6} + 14176 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 37384 T + 1240221324 T^{2} - 18715686161688 T^{3} + 444519628921633366 T^{4} - 18715686161688 p^{5} T^{5} + 1240221324 p^{10} T^{6} - 37384 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 11456 T + 1487883380 T^{2} - 10207240326912 T^{3} + 885850394336258646 T^{4} - 10207240326912 p^{5} T^{5} + 1487883380 p^{10} T^{6} + 11456 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 15752 T + 251537452 T^{2} + 18907846324056 T^{3} + 1392265203483209910 T^{4} + 18907846324056 p^{5} T^{5} + 251537452 p^{10} T^{6} + 15752 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 14032 T + 4075292836 T^{2} - 37348416655344 T^{3} + 7544579925679316358 T^{4} - 37348416655344 p^{5} T^{5} + 4075292836 p^{10} T^{6} - 14032 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 7552 T + 7061115716 T^{2} + 39517413965888 T^{3} + 18968128942170165926 T^{4} + 39517413965888 p^{5} T^{5} + 7061115716 p^{10} T^{6} + 7552 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 76104 T + 8970942652 T^{2} - 453081798431416 T^{3} + 28604127075938503974 T^{4} - 453081798431416 p^{5} T^{5} + 8970942652 p^{10} T^{6} - 76104 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 61056 T + 9407034300 T^{2} + 512811277886592 T^{3} + 40567621248160124998 T^{4} + 512811277886592 p^{5} T^{5} + 9407034300 p^{10} T^{6} + 61056 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 233856 T + 31172189780 T^{2} + 2964896486052928 T^{3} + \)\(21\!\cdots\!78\)\( T^{4} + 2964896486052928 p^{5} T^{5} + 31172189780 p^{10} T^{6} + 233856 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 86136 T + 15628720700 T^{2} - 1055365130959432 T^{3} + \)\(12\!\cdots\!98\)\( T^{4} - 1055365130959432 p^{5} T^{5} + 15628720700 p^{10} T^{6} - 86136 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 34664 T + 30099720028 T^{2} - 766306396866008 T^{3} + \)\(36\!\cdots\!18\)\( T^{4} - 766306396866008 p^{5} T^{5} + 30099720028 p^{10} T^{6} - 34664 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.41207973736681211725766963521, −7.03398761106698099477055053103, −6.60156698256176312986534596940, −6.54808797305051215125570994080, −6.51010176781368744890482943552, −5.93542562937579872222415942976, −5.84502561017796013158071244753, −5.60378161056747434002530798638, −5.45210237742259553200626864292, −5.24320583567421173657447011017, −4.79402840144759206561296732718, −4.74733949621629371010678213133, −4.59436972118887523517682051154, −4.18412220096567879177257220488, −3.94064024044333466250636926016, −3.70081002582961160690617050603, −3.69738870379290766701756430400, −2.76035962454376491693048972960, −2.62519220187063951380519140781, −2.31381533049260790471630204992, −2.14437592851221259758117633117, −1.51469152138437644897254001020, −1.47443454240362464698813516608, −1.04091468871633291076103873305, −1.02548064601404728500736125273, 0, 0, 0, 0, 1.02548064601404728500736125273, 1.04091468871633291076103873305, 1.47443454240362464698813516608, 1.51469152138437644897254001020, 2.14437592851221259758117633117, 2.31381533049260790471630204992, 2.62519220187063951380519140781, 2.76035962454376491693048972960, 3.69738870379290766701756430400, 3.70081002582961160690617050603, 3.94064024044333466250636926016, 4.18412220096567879177257220488, 4.59436972118887523517682051154, 4.74733949621629371010678213133, 4.79402840144759206561296732718, 5.24320583567421173657447011017, 5.45210237742259553200626864292, 5.60378161056747434002530798638, 5.84502561017796013158071244753, 5.93542562937579872222415942976, 6.51010176781368744890482943552, 6.54808797305051215125570994080, 6.60156698256176312986534596940, 7.03398761106698099477055053103, 7.41207973736681211725766963521

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