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Properties

Label 12-3808e6-1.1-c1e6-0-4
Degree $12$
Conductor $3.049\times 10^{21}$
Sign $1$
Analytic cond. $7.90395\times 10^{8}$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·5-s + 6·7-s + 3·9-s + 8·11-s + 8·13-s + 16·15-s − 6·17-s + 6·19-s + 24·21-s + 18·23-s − 25-s − 6·27-s − 8·29-s − 4·31-s + 32·33-s + 24·35-s − 8·37-s + 32·39-s + 16·41-s + 12·43-s + 12·45-s + 18·47-s + 21·49-s − 24·51-s − 2·53-s + 32·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 2.26·7-s + 9-s + 2.41·11-s + 2.21·13-s + 4.13·15-s − 1.45·17-s + 1.37·19-s + 5.23·21-s + 3.75·23-s − 1/5·25-s − 1.15·27-s − 1.48·29-s − 0.718·31-s + 5.57·33-s + 4.05·35-s − 1.31·37-s + 5.12·39-s + 2.49·41-s + 1.82·43-s + 1.78·45-s + 2.62·47-s + 3·49-s − 3.36·51-s − 0.274·53-s + 4.31·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(7.90395\times 10^{8}\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(192.8957297\)
\(L(\frac12)\) \(\approx\) \(192.8957297\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 - T )^{6} \)
17 \( ( 1 + T )^{6} \)
good3 \( 1 - 4 T + 13 T^{2} - 34 T^{3} + 26 p T^{4} - 164 T^{5} + 304 T^{6} - 164 p T^{7} + 26 p^{3} T^{8} - 34 p^{3} T^{9} + 13 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 4 T + 17 T^{2} - 38 T^{3} + 24 p T^{4} - 216 T^{5} + 616 T^{6} - 216 p T^{7} + 24 p^{3} T^{8} - 38 p^{3} T^{9} + 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 64 T^{2} - 316 T^{3} + 1571 T^{4} - 5772 T^{5} + 21704 T^{6} - 5772 p T^{7} + 1571 p^{2} T^{8} - 316 p^{3} T^{9} + 64 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 56 T^{2} - 252 T^{3} + 1139 T^{4} - 3820 T^{5} + 14824 T^{6} - 3820 p T^{7} + 1139 p^{2} T^{8} - 252 p^{3} T^{9} + 56 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 6 T + 106 T^{2} - 510 T^{3} + 4803 T^{4} - 18304 T^{5} + 119684 T^{6} - 18304 p T^{7} + 4803 p^{2} T^{8} - 510 p^{3} T^{9} + 106 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 18 T + 250 T^{2} - 102 p T^{3} + 18403 T^{4} - 114008 T^{5} + 606116 T^{6} - 114008 p T^{7} + 18403 p^{2} T^{8} - 102 p^{4} T^{9} + 250 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 8 T + 72 T^{2} + 500 T^{3} + 4203 T^{4} + 21724 T^{5} + 130584 T^{6} + 21724 p T^{7} + 4203 p^{2} T^{8} + 500 p^{3} T^{9} + 72 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 91 T^{2} + 386 T^{3} + 4658 T^{4} + 19242 T^{5} + 161892 T^{6} + 19242 p T^{7} + 4658 p^{2} T^{8} + 386 p^{3} T^{9} + 91 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 8 T + 170 T^{2} + 1148 T^{3} + 13347 T^{4} + 74676 T^{5} + 624612 T^{6} + 74676 p T^{7} + 13347 p^{2} T^{8} + 1148 p^{3} T^{9} + 170 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 16 T + 193 T^{2} - 1934 T^{3} + 18548 T^{4} - 136730 T^{5} + 924660 T^{6} - 136730 p T^{7} + 18548 p^{2} T^{8} - 1934 p^{3} T^{9} + 193 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 12 T + 141 T^{2} - 32 p T^{3} + 10410 T^{4} - 71640 T^{5} + 524920 T^{6} - 71640 p T^{7} + 10410 p^{2} T^{8} - 32 p^{4} T^{9} + 141 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 320 T^{2} - 3582 T^{3} + 37667 T^{4} - 6596 p T^{5} + 2354680 T^{6} - 6596 p^{2} T^{7} + 37667 p^{2} T^{8} - 3582 p^{3} T^{9} + 320 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 2 T + 157 T^{2} + 52 T^{3} + 14528 T^{4} + 2178 T^{5} + 17548 p T^{6} + 2178 p T^{7} + 14528 p^{2} T^{8} + 52 p^{3} T^{9} + 157 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 16 T + 358 T^{2} - 4048 T^{3} + 52039 T^{4} - 438144 T^{5} + 4059220 T^{6} - 438144 p T^{7} + 52039 p^{2} T^{8} - 4048 p^{3} T^{9} + 358 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T + 213 T^{2} - 750 T^{3} + 21092 T^{4} - 53490 T^{5} + 1483288 T^{6} - 53490 p T^{7} + 21092 p^{2} T^{8} - 750 p^{3} T^{9} + 213 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 247 T^{2} - 2840 T^{3} + 33674 T^{4} - 304768 T^{5} + 2894388 T^{6} - 304768 p T^{7} + 33674 p^{2} T^{8} - 2840 p^{3} T^{9} + 247 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 2 T + 150 T^{2} - 614 T^{3} + 12491 T^{4} - 73880 T^{5} + 1138268 T^{6} - 73880 p T^{7} + 12491 p^{2} T^{8} - 614 p^{3} T^{9} + 150 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 161 T^{2} - 1310 T^{3} + 11180 T^{4} - 115922 T^{5} + 573412 T^{6} - 115922 p T^{7} + 11180 p^{2} T^{8} - 1310 p^{3} T^{9} + 161 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 18 T + 288 T^{2} + 2302 T^{3} + 24499 T^{4} + 167772 T^{5} + 2003480 T^{6} + 167772 p T^{7} + 24499 p^{2} T^{8} + 2302 p^{3} T^{9} + 288 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 28 T + 624 T^{2} - 10128 T^{3} + 136411 T^{4} - 1560116 T^{5} + 15237096 T^{6} - 1560116 p T^{7} + 136411 p^{2} T^{8} - 10128 p^{3} T^{9} + 624 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T + 380 T^{2} + 530 T^{3} + 70659 T^{4} + 76140 T^{5} + 7900736 T^{6} + 76140 p T^{7} + 70659 p^{2} T^{8} + 530 p^{3} T^{9} + 380 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 18 T + 537 T^{2} + 7646 T^{3} + 124332 T^{4} + 1388888 T^{5} + 15835972 T^{6} + 1388888 p T^{7} + 124332 p^{2} T^{8} + 7646 p^{3} T^{9} + 537 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.11625244001601457786529108078, −4.10207269389219638472301980362, −4.06052625169988615320999152533, −3.88929291074873666133970829246, −3.87709771385512916219780849497, −3.65631802320396844894244554111, −3.57616133006817088611326683391, −3.27311350614320822700294330847, −3.16501878222752860108265975915, −2.98211709314446437782102382137, −2.82725595342919054570743658126, −2.75943651728509294832177774746, −2.68892715145109033232233148226, −2.34326434697725104640803437394, −2.08025920097999169492943297324, −2.07699634726073209072498801487, −1.97286296963843621883810294137, −1.86356518299678695951201277505, −1.71686010631985819566886202780, −1.46194624766015277001237384911, −1.16598987010097829758141949310, −0.978888938983805615210256949315, −0.850227496460415026977711207811, −0.73022592794415503579622005126, −0.66353426530292426963187070118, 0.66353426530292426963187070118, 0.73022592794415503579622005126, 0.850227496460415026977711207811, 0.978888938983805615210256949315, 1.16598987010097829758141949310, 1.46194624766015277001237384911, 1.71686010631985819566886202780, 1.86356518299678695951201277505, 1.97286296963843621883810294137, 2.07699634726073209072498801487, 2.08025920097999169492943297324, 2.34326434697725104640803437394, 2.68892715145109033232233148226, 2.75943651728509294832177774746, 2.82725595342919054570743658126, 2.98211709314446437782102382137, 3.16501878222752860108265975915, 3.27311350614320822700294330847, 3.57616133006817088611326683391, 3.65631802320396844894244554111, 3.87709771385512916219780849497, 3.88929291074873666133970829246, 4.06052625169988615320999152533, 4.10207269389219638472301980362, 4.11625244001601457786529108078

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

Plot not available for L-functions of degree greater than 10.