Properties

Label 4-1064e2-1.1-c1e2-0-9
Degree $4$
Conductor $1132096$
Sign $1$
Analytic cond. $72.1834$
Root an. cond. $2.91480$
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  − 2·3-s − 5-s − 5·7-s + 3·9-s + 3·11-s + 8·13-s + 2·15-s − 2·17-s − 19-s + 10·21-s + 7·23-s + 5·25-s − 10·27-s + 4·29-s − 6·31-s − 6·33-s + 5·35-s + 10·37-s − 16·39-s + 16·41-s + 14·43-s − 3·45-s − 9·47-s + 18·49-s + 4·51-s − 6·53-s − 3·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.88·7-s + 9-s + 0.904·11-s + 2.21·13-s + 0.516·15-s − 0.485·17-s − 0.229·19-s + 2.18·21-s + 1.45·23-s + 25-s − 1.92·27-s + 0.742·29-s − 1.07·31-s − 1.04·33-s + 0.845·35-s + 1.64·37-s − 2.56·39-s + 2.49·41-s + 2.13·43-s − 0.447·45-s − 1.31·47-s + 18/7·49-s + 0.560·51-s − 0.824·53-s − 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1132096\)    =    \(2^{6} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(72.1834\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1064} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1132096,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.245844969\)
\(L(\frac12)\) \(\approx\) \(1.245844969\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
19$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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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

−10.16520660527086228687818176258, −9.542237632783190363110083671897, −9.203921037253023897027227078430, −9.096827297599335463506542525907, −8.728276234555995029550995546085, −7.81591177673674477281837322463, −7.55755532942875722679776167276, −7.07209841530330156611002946338, −6.39389551720631166389588537865, −6.38377505530515802809230800678, −5.89564055134597059218867911319, −5.84008536742768941390217106627, −4.73060125098432165301508575413, −4.45532640588309260919232421281, −3.91975547138372311145867777825, −3.36151952398278646930131537947, −3.14971981481030273137137794166, −2.14700054113443752242126418183, −1.04145490339913171875300757118, −0.70221079048893333270470609790, 0.70221079048893333270470609790, 1.04145490339913171875300757118, 2.14700054113443752242126418183, 3.14971981481030273137137794166, 3.36151952398278646930131537947, 3.91975547138372311145867777825, 4.45532640588309260919232421281, 4.73060125098432165301508575413, 5.84008536742768941390217106627, 5.89564055134597059218867911319, 6.38377505530515802809230800678, 6.39389551720631166389588537865, 7.07209841530330156611002946338, 7.55755532942875722679776167276, 7.81591177673674477281837322463, 8.728276234555995029550995546085, 9.096827297599335463506542525907, 9.203921037253023897027227078430, 9.542237632783190363110083671897, 10.16520660527086228687818176258

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