Properties

Label 4-3626e2-1.1-c1e2-0-20
Degree $4$
Conductor $13147876$
Sign $1$
Analytic cond. $838.319$
Root an. cond. $5.38086$
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 + 3·4-s + 4·8-s − 6·9-s + 5·16-s − 12·18-s − 12·23-s − 10·25-s − 12·29-s + 6·32-s − 18·36-s + 2·37-s − 8·43-s − 24·46-s − 20·50-s − 24·58-s + 7·64-s − 16·67-s − 24·72-s + 4·74-s − 20·79-s + 27·81-s − 16·86-s − 36·92-s − 30·100-s − 8·107-s − 20·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s + 5/4·16-s − 2.82·18-s − 2.50·23-s − 2·25-s − 2.22·29-s + 1.06·32-s − 3·36-s + 0.328·37-s − 1.21·43-s − 3.53·46-s − 2.82·50-s − 3.15·58-s + 7/8·64-s − 1.95·67-s − 2.82·72-s + 0.464·74-s − 2.25·79-s + 3·81-s − 1.72·86-s − 3.75·92-s − 3·100-s − 0.773·107-s − 1.91·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13147876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13147876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13147876\)    =    \(2^{2} \cdot 7^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(838.319\)
Root analytic conductor: \(5.38086\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13147876,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 170 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.135690249933689692712030383394, −7.952932978275074682316466454396, −7.51467785242406479008944379454, −7.44341221500459362652491884781, −6.47407787691715178504049915516, −6.43174950881267442675402629657, −5.87876252389419456870462642474, −5.80867489247740207901158607553, −5.28990484688904571809568106469, −5.25774638355562507268228364354, −4.24413829548230577512165382412, −4.23546773353991904414238491024, −3.56095299368233930503112516588, −3.54452487667790804262588064127, −2.62938535497031803619662259325, −2.61937931691366260674987407754, −1.73254520447919951104038560393, −1.72911612108547849255571744329, 0, 0, 1.72911612108547849255571744329, 1.73254520447919951104038560393, 2.61937931691366260674987407754, 2.62938535497031803619662259325, 3.54452487667790804262588064127, 3.56095299368233930503112516588, 4.23546773353991904414238491024, 4.24413829548230577512165382412, 5.25774638355562507268228364354, 5.28990484688904571809568106469, 5.80867489247740207901158607553, 5.87876252389419456870462642474, 6.43174950881267442675402629657, 6.47407787691715178504049915516, 7.44341221500459362652491884781, 7.51467785242406479008944379454, 7.952932978275074682316466454396, 8.135690249933689692712030383394

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