Properties

Label 4-8470e2-1.1-c1e2-0-16
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
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 + 3·4-s − 2·5-s − 2·7-s + 4·8-s − 9-s − 4·10-s − 2·13-s − 4·14-s + 5·16-s − 2·17-s − 2·18-s − 6·20-s − 12·23-s + 3·25-s − 4·26-s − 6·28-s − 6·31-s + 6·32-s − 4·34-s + 4·35-s − 3·36-s − 8·40-s + 10·41-s − 2·43-s + 2·45-s − 24·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s − 1/3·9-s − 1.26·10-s − 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.485·17-s − 0.471·18-s − 1.34·20-s − 2.50·23-s + 3/5·25-s − 0.784·26-s − 1.13·28-s − 1.07·31-s + 1.06·32-s − 0.685·34-s + 0.676·35-s − 1/2·36-s − 1.26·40-s + 1.56·41-s − 0.304·43-s + 0.298·45-s − 3.53·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
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: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 77 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 217 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 165 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−7.45056925512070115416987262798, −7.33217108056062544220941083358, −6.81536159609898956893276444539, −6.56680701793075694922079342197, −6.12564085382565302356541290916, −5.81716842022777483780480641274, −5.57310032861518723656973116922, −5.26962666196110380987285246569, −4.66743675558513689464816698782, −4.26198358683539110006474451785, −4.06758890402606218017066074910, −3.93801086902293806633427444106, −3.26264498459055322419460456194, −3.08794132767576732838835069370, −2.45912566335693024872813918286, −2.31986173042968351033329452035, −1.72002553879532151432746470475, −1.07933078990502846177202536579, 0, 0, 1.07933078990502846177202536579, 1.72002553879532151432746470475, 2.31986173042968351033329452035, 2.45912566335693024872813918286, 3.08794132767576732838835069370, 3.26264498459055322419460456194, 3.93801086902293806633427444106, 4.06758890402606218017066074910, 4.26198358683539110006474451785, 4.66743675558513689464816698782, 5.26962666196110380987285246569, 5.57310032861518723656973116922, 5.81716842022777483780480641274, 6.12564085382565302356541290916, 6.56680701793075694922079342197, 6.81536159609898956893276444539, 7.33217108056062544220941083358, 7.45056925512070115416987262798

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