Properties

Label 4-5610e2-1.1-c1e2-0-4
Degree $4$
Conductor $31472100$
Sign $1$
Analytic cond. $2006.68$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s − 6·12-s + 2·13-s + 4·14-s + 4·15-s + 5·16-s − 2·17-s − 6·18-s − 2·19-s − 6·20-s + 4·21-s + 4·22-s + 2·23-s + 8·24-s + 3·25-s − 4·26-s − 4·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s − 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31472100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2006.68\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 31472100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_4$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_4$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.72579528193371846491705213349, −7.69282559511759486298463732786, −7.42742845960879789994159223270, −6.93941952073029132568959007777, −6.52161677746673874930792949851, −6.32504887213968881735474383656, −5.95847921420948554713859963582, −5.72001551675743498886300221038, −4.98629576348545896343190146445, −4.85278165459810449003758763802, −4.09793630490715019314108975147, −4.06853825893635969694119455988, −3.31958827164474444638377309570, −2.97532646919772594957301820161, −2.53261331735217934787494582682, −1.92339056725753052652835142211, −1.30233501805382164955410724687, −0.849150571873832115931609819761, 0, 0, 0.849150571873832115931609819761, 1.30233501805382164955410724687, 1.92339056725753052652835142211, 2.53261331735217934787494582682, 2.97532646919772594957301820161, 3.31958827164474444638377309570, 4.06853825893635969694119455988, 4.09793630490715019314108975147, 4.85278165459810449003758763802, 4.98629576348545896343190146445, 5.72001551675743498886300221038, 5.95847921420948554713859963582, 6.32504887213968881735474383656, 6.52161677746673874930792949851, 6.93941952073029132568959007777, 7.42742845960879789994159223270, 7.69282559511759486298463732786, 7.72579528193371846491705213349

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