Properties

Label 4-637e2-1.1-c1e2-0-24
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 8·8-s − 6·9-s − 6·11-s − 7·16-s − 12·18-s − 12·22-s − 12·23-s − 10·25-s − 10·29-s + 14·32-s + 6·36-s + 16·37-s + 4·43-s + 6·44-s − 24·46-s − 20·50-s − 6·53-s − 20·58-s + 35·64-s − 6·67-s − 10·71-s + 48·72-s + 32·74-s − 12·79-s + 27·81-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2·9-s − 1.80·11-s − 7/4·16-s − 2.82·18-s − 2.55·22-s − 2.50·23-s − 2·25-s − 1.85·29-s + 2.47·32-s + 36-s + 2.63·37-s + 0.609·43-s + 0.904·44-s − 3.53·46-s − 2.82·50-s − 0.824·53-s − 2.62·58-s + 35/8·64-s − 0.733·67-s − 1.18·71-s + 5.65·72-s + 3.71·74-s − 1.35·79-s + 3·81-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7 \( 1 \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.264528127864852069730211246497, −7.71886042427351779334180674638, −7.58221538537525761723731254609, −6.13619288958387853257590270753, −5.97764942510297289458867575451, −5.71244157014039939594521038014, −5.56220221892590651265131476874, −4.57403153778540150751986491981, −4.53493125422219918903828313314, −3.54025539217060415386292648543, −3.52729931218747647603184716659, −2.43258577292117985821674468072, −2.43005796671753573111852467369, 0, 0, 2.43005796671753573111852467369, 2.43258577292117985821674468072, 3.52729931218747647603184716659, 3.54025539217060415386292648543, 4.53493125422219918903828313314, 4.57403153778540150751986491981, 5.56220221892590651265131476874, 5.71244157014039939594521038014, 5.97764942510297289458867575451, 6.13619288958387853257590270753, 7.58221538537525761723731254609, 7.71886042427351779334180674638, 8.264528127864852069730211246497

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