Properties

Label 4-637e2-1.1-c1e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 3·5-s + 3·9-s − 9·11-s − 12-s + 2·13-s + 3·15-s − 3·16-s − 12·17-s + 3·19-s − 3·20-s + 25-s − 8·27-s − 3·29-s + 3·31-s + 9·33-s + 3·36-s − 2·39-s + 9·41-s − 11·43-s − 9·44-s − 9·45-s − 15·47-s + 3·48-s + 12·51-s + 2·52-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 1.34·5-s + 9-s − 2.71·11-s − 0.288·12-s + 0.554·13-s + 0.774·15-s − 3/4·16-s − 2.91·17-s + 0.688·19-s − 0.670·20-s + 1/5·25-s − 1.53·27-s − 0.557·29-s + 0.538·31-s + 1.56·33-s + 1/2·36-s − 0.320·39-s + 1.40·41-s − 1.67·43-s − 1.35·44-s − 1.34·45-s − 2.18·47-s + 0.433·48-s + 1.68·51-s + 0.277·52-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

\(L(1)\) \(\approx\) \(0.2279956952\)
\(L(\frac12)\) \(\approx\) \(0.2279956952\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 15 T + 122 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−11.22783433584987940958054786872, −10.37087380128518449082953256803, −10.24681892052838759436974869280, −9.485150478624857185707624148994, −9.134862506305999947036118212250, −8.297475241199766100435902232246, −8.187769139952625408806874395660, −7.72100756015289648089441662630, −7.27285222135996009362554055102, −6.75416339491388564675262158490, −6.59635508802287782532496163457, −5.77607764717510825153228139047, −5.18780692382171607647907423357, −4.87230657305329020076787892974, −4.19914138268499238712459652604, −3.95933413418475588613507594495, −3.02418231469773764169373465398, −2.40279497122692555087689854232, −1.87852419102190085795320780970, −0.25148432323485417672786791210, 0.25148432323485417672786791210, 1.87852419102190085795320780970, 2.40279497122692555087689854232, 3.02418231469773764169373465398, 3.95933413418475588613507594495, 4.19914138268499238712459652604, 4.87230657305329020076787892974, 5.18780692382171607647907423357, 5.77607764717510825153228139047, 6.59635508802287782532496163457, 6.75416339491388564675262158490, 7.27285222135996009362554055102, 7.72100756015289648089441662630, 8.187769139952625408806874395660, 8.297475241199766100435902232246, 9.134862506305999947036118212250, 9.485150478624857185707624148994, 10.24681892052838759436974869280, 10.37087380128518449082953256803, 11.22783433584987940958054786872

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