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Properties

Label 12-570e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  + 3·7-s + 8-s − 3·11-s − 15·13-s + 9·17-s + 9·19-s + 15·23-s − 27-s + 3·29-s − 18·31-s − 42·37-s − 12·41-s − 15·43-s − 24·47-s + 21·49-s − 18·53-s + 3·56-s + 3·59-s + 21·61-s + 30·67-s + 42·71-s − 3·73-s − 9·77-s − 39·79-s + 15·83-s − 3·88-s + 3·89-s + ⋯
L(s)  = 1  + 1.13·7-s + 0.353·8-s − 0.904·11-s − 4.16·13-s + 2.18·17-s + 2.06·19-s + 3.12·23-s − 0.192·27-s + 0.557·29-s − 3.23·31-s − 6.90·37-s − 1.87·41-s − 2.28·43-s − 3.50·47-s + 3·49-s − 2.47·53-s + 0.400·56-s + 0.390·59-s + 2.68·61-s + 3.66·67-s + 4.98·71-s − 0.351·73-s − 1.02·77-s − 4.38·79-s + 1.64·83-s − 0.319·88-s + 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.0005141098667\)
\(L(\frac12)\) \(\approx\) \(0.0005141098667\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 + T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( 1 - 9 T + 179 T^{3} - 9 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 18 p T^{5} - 1161 T^{6} - 18 p^{2} T^{7} + 171 p^{2} T^{8} + 19 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 6 T^{2} + 27 T^{3} + 3 p T^{4} - 492 T^{5} - 821 T^{6} - 492 p T^{7} + 3 p^{3} T^{8} + 27 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 15 T + 120 T^{2} + 698 T^{3} + 3303 T^{4} + 13401 T^{5} + 49605 T^{6} + 13401 p T^{7} + 3303 p^{2} T^{8} + 698 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 9 T + 18 T^{2} + 130 T^{3} - 747 T^{4} - 27 p T^{5} + 12581 T^{6} - 27 p^{2} T^{7} - 747 p^{2} T^{8} + 130 p^{3} T^{9} + 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 81 T^{2} - 63 T^{3} - 522 T^{4} - 6450 T^{5} + 65881 T^{6} - 6450 p T^{7} - 522 p^{2} T^{8} - 63 p^{3} T^{9} + 81 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 3 T + 15 T^{2} - 35 T^{3} + 708 T^{4} + 36 p T^{5} - 15667 T^{6} + 36 p^{2} T^{7} + 708 p^{2} T^{8} - 35 p^{3} T^{9} + 15 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 18 T + 132 T^{2} + 882 T^{3} + 7746 T^{4} + 50130 T^{5} + 261803 T^{6} + 50130 p T^{7} + 7746 p^{2} T^{8} + 882 p^{3} T^{9} + 132 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 21 T + 231 T^{2} + 1681 T^{3} + 231 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 12 T + 42 T^{2} - 145 T^{3} - 1227 T^{4} + 5265 T^{5} + 97193 T^{6} + 5265 p T^{7} - 1227 p^{2} T^{8} - 145 p^{3} T^{9} + 42 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 12 T^{2} - 1174 T^{3} - 7488 T^{4} + 26289 T^{5} + 487041 T^{6} + 26289 p T^{7} - 7488 p^{2} T^{8} - 1174 p^{3} T^{9} + 12 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 24 T + 396 T^{2} + 4905 T^{3} + 49761 T^{4} + 431367 T^{5} + 3169513 T^{6} + 431367 p T^{7} + 49761 p^{2} T^{8} + 4905 p^{3} T^{9} + 396 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 135 T^{2} + 513 T^{3} - 459 T^{4} - 37431 T^{5} - 414314 T^{6} - 37431 p T^{7} - 459 p^{2} T^{8} + 513 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T + 48 T^{2} - 988 T^{3} + 4998 T^{4} - 36675 T^{5} + 684653 T^{6} - 36675 p T^{7} + 4998 p^{2} T^{8} - 988 p^{3} T^{9} + 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 21 T + 396 T^{2} - 5376 T^{3} + 61551 T^{4} - 606783 T^{5} + 5015825 T^{6} - 606783 p T^{7} + 61551 p^{2} T^{8} - 5376 p^{3} T^{9} + 396 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 30 T + 432 T^{2} - 4017 T^{3} + 32589 T^{4} - 299091 T^{5} + 2682629 T^{6} - 299091 p T^{7} + 32589 p^{2} T^{8} - 4017 p^{3} T^{9} + 432 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 42 T + 756 T^{2} - 7317 T^{3} + 30933 T^{4} + 161049 T^{5} - 3121451 T^{6} + 161049 p T^{7} + 30933 p^{2} T^{8} - 7317 p^{3} T^{9} + 756 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 3 T + 246 T^{2} + 220 T^{3} + 30753 T^{4} - 9423 T^{5} + 2559561 T^{6} - 9423 p T^{7} + 30753 p^{2} T^{8} + 220 p^{3} T^{9} + 246 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 39 T + 621 T^{2} + 4539 T^{3} - 4374 T^{4} - 507534 T^{5} - 6364783 T^{6} - 507534 p T^{7} - 4374 p^{2} T^{8} + 4539 p^{3} T^{9} + 621 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 15 T + 300 T^{2} - 3717 T^{3} + 48165 T^{4} - 471660 T^{5} + 4987915 T^{6} - 471660 p T^{7} + 48165 p^{2} T^{8} - 3717 p^{3} T^{9} + 300 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T - 12 T^{2} - 620 T^{3} - 4242 T^{4} + 92853 T^{5} - 42937 T^{6} + 92853 p T^{7} - 4242 p^{2} T^{8} - 620 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 18 T + 54 T^{2} - 1870 T^{3} - 21636 T^{4} + 36360 T^{5} + 2056107 T^{6} + 36360 p T^{7} - 21636 p^{2} T^{8} - 1870 p^{3} T^{9} + 54 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.56811787935747050009170629895, −5.55963360788584169019642869555, −5.18795135548182688485425313032, −5.15264377376796712554147751317, −5.10944609413149526633166910534, −5.02939841881568695528218063571, −4.96676712884316885146914532081, −4.76698957279706603935881082885, −4.59771913846079886700333637162, −4.30537436019756683834860160925, −3.64157685142095660183382236293, −3.59538493517752789035169621668, −3.51384634605734658405478140339, −3.32507181093303288893358341960, −3.30996022686348678100707883742, −3.19230474073867697764224246501, −2.68866254669931057061051355636, −2.56367965758591254670156197830, −2.04360545762475382680193357773, −1.91264588952327387506524995440, −1.87816244016718787915792965853, −1.74049952061666223572130267527, −1.14787200568262356431535023228, −0.873891891203579807056629521243, −0.00477455418305308418209864698, 0.00477455418305308418209864698, 0.873891891203579807056629521243, 1.14787200568262356431535023228, 1.74049952061666223572130267527, 1.87816244016718787915792965853, 1.91264588952327387506524995440, 2.04360545762475382680193357773, 2.56367965758591254670156197830, 2.68866254669931057061051355636, 3.19230474073867697764224246501, 3.30996022686348678100707883742, 3.32507181093303288893358341960, 3.51384634605734658405478140339, 3.59538493517752789035169621668, 3.64157685142095660183382236293, 4.30537436019756683834860160925, 4.59771913846079886700333637162, 4.76698957279706603935881082885, 4.96676712884316885146914532081, 5.02939841881568695528218063571, 5.10944609413149526633166910534, 5.15264377376796712554147751317, 5.18795135548182688485425313032, 5.55963360788584169019642869555, 5.56811787935747050009170629895

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

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