Properties

Label 4-637e2-1.1-c1e2-0-10
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $25.8721$
Root an. cond. $2.25532$
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  + 2·2-s + 2·4-s − 3·5-s + 4·8-s + 3·9-s − 6·10-s + 6·11-s + 2·13-s + 8·16-s + 4·17-s + 6·18-s + 5·19-s − 6·20-s + 12·22-s − 3·23-s + 5·25-s + 4·26-s − 10·29-s − 3·31-s + 8·32-s + 8·34-s + 6·36-s + 4·37-s + 10·38-s − 12·40-s + 12·41-s − 2·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.34·5-s + 1.41·8-s + 9-s − 1.89·10-s + 1.80·11-s + 0.554·13-s + 2·16-s + 0.970·17-s + 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s − 0.625·23-s + 25-s + 0.784·26-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 1.37·34-s + 36-s + 0.657·37-s + 1.62·38-s − 1.89·40-s + 1.87·41-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(25.8721\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.776777761\)
\(L(\frac12)\) \(\approx\) \(4.776777761\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−10.85136710588636109608076253036, −10.65692092460855196745324440424, −9.913449355444814881784477685814, −9.568364925883983481630577943600, −9.167391887771824174634402017319, −8.470982317285759574967752985939, −8.032324588200659913296266724006, −7.35196007869106240365006618850, −7.18908742508590601313968260101, −7.15366505102637705856941077257, −5.99197996786868933247458472617, −5.81999417995825630018858231590, −5.31296843448578688893409175778, −4.39517148428385295574554083289, −4.20341065449296861761035057529, −3.90337222200630746688036814772, −3.57050520560488258897162781027, −2.80261427554416638192209594583, −1.50938155335232479980636395902, −1.20919581381145784545195474837, 1.20919581381145784545195474837, 1.50938155335232479980636395902, 2.80261427554416638192209594583, 3.57050520560488258897162781027, 3.90337222200630746688036814772, 4.20341065449296861761035057529, 4.39517148428385295574554083289, 5.31296843448578688893409175778, 5.81999417995825630018858231590, 5.99197996786868933247458472617, 7.15366505102637705856941077257, 7.18908742508590601313968260101, 7.35196007869106240365006618850, 8.032324588200659913296266724006, 8.470982317285759574967752985939, 9.167391887771824174634402017319, 9.568364925883983481630577943600, 9.913449355444814881784477685814, 10.65692092460855196745324440424, 10.85136710588636109608076253036

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