Properties

Label 12-570e6-1.1-c1e6-0-4
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 3·9-s + 16·11-s + 6·16-s − 6·19-s + 25-s − 20·29-s + 9·36-s + 8·41-s − 48·44-s + 10·49-s − 20·59-s + 28·61-s − 10·64-s + 40·71-s + 18·76-s + 6·81-s − 48·89-s − 48·99-s − 3·100-s + 80·101-s − 44·109-s + 60·116-s + 110·121-s + 16·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 3/2·4-s − 9-s + 4.82·11-s + 3/2·16-s − 1.37·19-s + 1/5·25-s − 3.71·29-s + 3/2·36-s + 1.24·41-s − 7.23·44-s + 10/7·49-s − 2.60·59-s + 3.58·61-s − 5/4·64-s + 4.74·71-s + 2.06·76-s + 2/3·81-s − 5.08·89-s − 4.82·99-s − 0.299·100-s + 7.96·101-s − 4.21·109-s + 5.57·116-s + 10·121-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(2.189543038\)
\(L(\frac12)\) \(\approx\) \(2.189543038\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \)
19 \( ( 1 + T )^{6} \)
good7 \( 1 - 10 T^{2} + 95 T^{4} - 780 T^{6} + 95 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 8 T + 41 T^{2} - 160 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 30 T^{2} + 551 T^{4} - 7556 T^{6} + 551 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 58 T^{2} + 1775 T^{4} - 36588 T^{6} + 1775 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 90 T^{2} + 3839 T^{4} - 105452 T^{6} + 3839 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 10 T + 107 T^{2} + 572 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} )^{2} \)
37 \( 1 - 174 T^{2} + 13943 T^{4} - 655652 T^{6} + 13943 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 4 T + 43 T^{2} + 72 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 26 T^{2} + 3383 T^{4} + 31500 T^{6} + 3383 p^{2} T^{8} + 26 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 234 T^{2} + 24431 T^{4} - 1470092 T^{6} + 24431 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{3} \)
59 \( ( 1 + 10 T + 173 T^{2} + 1172 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 98 T^{2} + 967 T^{4} + 251140 T^{6} + 967 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 20 T + 261 T^{2} - 2520 T^{3} + 261 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 246 T^{2} + 32063 T^{4} - 2771828 T^{6} + 32063 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 221 T^{2} + 16 T^{3} + 221 p T^{4} + p^{3} T^{6} )^{2} \)
83 \( 1 - 130 T^{2} + 13415 T^{4} - 1553724 T^{6} + 13415 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 24 T + 443 T^{2} + 4672 T^{3} + 443 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 394 T^{2} + 77071 T^{4} - 9275212 T^{6} + 77071 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.79392936258933716759737369588, −5.56564875337071980002908405314, −5.30668907910744466780797887346, −5.30428127178339091425732015310, −5.24126470178322261892898971211, −4.91069135163245847719790275573, −4.56421428813082819501461395220, −4.29222411480810961400181157066, −4.27377215473536714194381455765, −4.25119854084832143991695701691, −3.88730176969095619246564633366, −3.84268928501441211330844908388, −3.71221800076445773810352542758, −3.49289241883032846454710924905, −3.43705856575655104629233546405, −3.18146253185048356371363834082, −2.62905359638996164785532931403, −2.50701061230536849326127293194, −2.24940541001662021368960930817, −1.85999574110704942878460076210, −1.73330909398726082051822632408, −1.53183109142560054556249760847, −0.992403568753955883831446986520, −0.883583711262000729536427883987, −0.38826299121997028302726111760, 0.38826299121997028302726111760, 0.883583711262000729536427883987, 0.992403568753955883831446986520, 1.53183109142560054556249760847, 1.73330909398726082051822632408, 1.85999574110704942878460076210, 2.24940541001662021368960930817, 2.50701061230536849326127293194, 2.62905359638996164785532931403, 3.18146253185048356371363834082, 3.43705856575655104629233546405, 3.49289241883032846454710924905, 3.71221800076445773810352542758, 3.84268928501441211330844908388, 3.88730176969095619246564633366, 4.25119854084832143991695701691, 4.27377215473536714194381455765, 4.29222411480810961400181157066, 4.56421428813082819501461395220, 4.91069135163245847719790275573, 5.24126470178322261892898971211, 5.30428127178339091425732015310, 5.30668907910744466780797887346, 5.56564875337071980002908405314, 5.79392936258933716759737369588

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

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