Properties

Label 12-8470e6-1.1-c1e6-0-4
Degree $12$
Conductor $3.692\times 10^{23}$
Sign $1$
Analytic cond. $9.57112\times 10^{10}$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 21·4-s − 6·5-s + 6·7-s − 56·8-s + 36·10-s − 36·14-s + 126·16-s + 6·17-s − 126·20-s + 21·25-s − 4·27-s + 126·28-s + 12·29-s − 252·32-s − 36·34-s − 36·35-s + 24·37-s + 336·40-s + 12·41-s − 18·43-s + 24·47-s + 21·49-s − 126·50-s + 36·53-s + 24·54-s − 336·56-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s − 2.68·5-s + 2.26·7-s − 19.7·8-s + 11.3·10-s − 9.62·14-s + 63/2·16-s + 1.45·17-s − 28.1·20-s + 21/5·25-s − 0.769·27-s + 23.8·28-s + 2.22·29-s − 44.5·32-s − 6.17·34-s − 6.08·35-s + 3.94·37-s + 53.1·40-s + 1.87·41-s − 2.74·43-s + 3.50·47-s + 3·49-s − 17.8·50-s + 4.94·53-s + 3.26·54-s − 44.8·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.57112\times 10^{10}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.419760169\)
\(L(\frac12)\) \(\approx\) \(5.419760169\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
7 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( 1 + 4 T^{3} + 10 T^{6} + 4 p^{3} T^{9} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} - 8 T^{3} + 792 T^{4} - 144 T^{5} + 11862 T^{6} - 144 p T^{7} + 792 p^{2} T^{8} - 8 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 6 T + 3 p T^{2} - 158 T^{3} + 819 T^{4} - 804 T^{5} + 7478 T^{6} - 804 p T^{7} + 819 p^{2} T^{8} - 158 p^{3} T^{9} + 3 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 69 T^{2} - 104 T^{3} + 2094 T^{4} - 5712 T^{5} + 43329 T^{6} - 5712 p T^{7} + 2094 p^{2} T^{8} - 104 p^{3} T^{9} + 69 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 + 84 T^{2} + 72 T^{3} + 3696 T^{4} + 3024 T^{5} + 106666 T^{6} + 3024 p T^{7} + 3696 p^{2} T^{8} + 72 p^{3} T^{9} + 84 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 12 T + 162 T^{2} - 1292 T^{3} + 10407 T^{4} - 62904 T^{5} + 381500 T^{6} - 62904 p T^{7} + 10407 p^{2} T^{8} - 1292 p^{3} T^{9} + 162 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 78 T^{2} + 256 T^{3} + 3507 T^{4} + 12288 T^{5} + 140412 T^{6} + 12288 p T^{7} + 3507 p^{2} T^{8} + 256 p^{3} T^{9} + 78 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 24 T + 390 T^{2} - 4600 T^{3} + 43575 T^{4} - 342480 T^{5} + 2259060 T^{6} - 342480 p T^{7} + 43575 p^{2} T^{8} - 4600 p^{3} T^{9} + 390 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 12 T + 138 T^{2} - 924 T^{3} + 7683 T^{4} - 36552 T^{5} + 269044 T^{6} - 36552 p T^{7} + 7683 p^{2} T^{8} - 924 p^{3} T^{9} + 138 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 339 T^{2} + 3758 T^{3} + 40299 T^{4} + 315996 T^{5} + 2370054 T^{6} + 315996 p T^{7} + 40299 p^{2} T^{8} + 3758 p^{3} T^{9} + 339 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 24 T + 426 T^{2} - 5160 T^{3} + 53907 T^{4} - 452976 T^{5} + 3408532 T^{6} - 452976 p T^{7} + 53907 p^{2} T^{8} - 5160 p^{3} T^{9} + 426 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 36 T + 723 T^{2} - 10292 T^{3} + 116115 T^{4} - 1088040 T^{5} + 8606582 T^{6} - 1088040 p T^{7} + 116115 p^{2} T^{8} - 10292 p^{3} T^{9} + 723 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 30 T + 525 T^{2} - 6326 T^{3} + 62298 T^{4} - 530430 T^{5} + 4233821 T^{6} - 530430 p T^{7} + 62298 p^{2} T^{8} - 6326 p^{3} T^{9} + 525 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 36 T + 783 T^{2} - 11852 T^{3} + 141867 T^{4} - 1391616 T^{5} + 11736378 T^{6} - 1391616 p T^{7} + 141867 p^{2} T^{8} - 11852 p^{3} T^{9} + 783 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 12 T + 255 T^{2} + 1268 T^{3} + 13731 T^{4} - 44472 T^{5} + 129678 T^{6} - 44472 p T^{7} + 13731 p^{2} T^{8} + 1268 p^{3} T^{9} + 255 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T + 315 T^{2} - 1930 T^{3} + 47739 T^{4} - 254292 T^{5} + 4315922 T^{6} - 254292 p T^{7} + 47739 p^{2} T^{8} - 1930 p^{3} T^{9} + 315 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T + 267 T^{2} + 846 T^{3} + 31803 T^{4} + 43332 T^{5} + 34886 p T^{6} + 43332 p T^{7} + 31803 p^{2} T^{8} + 846 p^{3} T^{9} + 267 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 549 T^{2} + 8120 T^{3} + 109554 T^{4} + 1169016 T^{5} + 11455461 T^{6} + 1169016 p T^{7} + 109554 p^{2} T^{8} + 8120 p^{3} T^{9} + 549 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 24 T + 576 T^{2} - 7528 T^{3} + 101928 T^{4} - 942792 T^{5} + 9994598 T^{6} - 942792 p T^{7} + 101928 p^{2} T^{8} - 7528 p^{3} T^{9} + 576 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 36 T + 858 T^{2} - 15044 T^{3} + 216771 T^{4} - 2593992 T^{5} + 26619668 T^{6} - 2593992 p T^{7} + 216771 p^{2} T^{8} - 15044 p^{3} T^{9} + 858 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 195 T^{2} - 536 T^{3} + 21363 T^{4} - 59496 T^{5} + 2334450 T^{6} - 59496 p T^{7} + 21363 p^{2} T^{8} - 536 p^{3} T^{9} + 195 p^{4} T^{10} + p^{6} T^{12} \)
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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

−3.94258648487291169376190434606, −3.86693714984605330692601325761, −3.48170968142739425433466725231, −3.45425326130591218365293145911, −3.33422129339134223644718132628, −3.27532200538429636127554623235, −3.25507325687835547392654169996, −2.58485336507266004238571748744, −2.57945538919811349799486136995, −2.54446588906533650367396748174, −2.52486566426466165061235032103, −2.51626716325449937032407626288, −2.31569445202895991463566412741, −1.97760172332142546599807906028, −1.88346431349118590049539035192, −1.85108613436538160961602832999, −1.49682365410774898865416167571, −1.23426748230135058269384312715, −1.17646842276301766472017456730, −0.877558692510488124636777721113, −0.73843343586857687839866900778, −0.65821886850310201935194505795, −0.61761996794550648831866080738, −0.59683665396852459685297512396, −0.57424432940998752590300254526, 0.57424432940998752590300254526, 0.59683665396852459685297512396, 0.61761996794550648831866080738, 0.65821886850310201935194505795, 0.73843343586857687839866900778, 0.877558692510488124636777721113, 1.17646842276301766472017456730, 1.23426748230135058269384312715, 1.49682365410774898865416167571, 1.85108613436538160961602832999, 1.88346431349118590049539035192, 1.97760172332142546599807906028, 2.31569445202895991463566412741, 2.51626716325449937032407626288, 2.52486566426466165061235032103, 2.54446588906533650367396748174, 2.57945538919811349799486136995, 2.58485336507266004238571748744, 3.25507325687835547392654169996, 3.27532200538429636127554623235, 3.33422129339134223644718132628, 3.45425326130591218365293145911, 3.48170968142739425433466725231, 3.86693714984605330692601325761, 3.94258648487291169376190434606

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

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