Properties

Label 4-260e2-1.1-c1e2-0-11
Degree $4$
Conductor $67600$
Sign $1$
Analytic cond. $4.31023$
Root an. cond. $1.44087$
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  + 3·3-s + 2·5-s − 3·7-s + 3·9-s − 3·11-s + 2·13-s + 6·15-s + 7·17-s − 19-s − 9·21-s + 7·23-s + 3·25-s + 5·29-s − 8·31-s − 9·33-s − 6·35-s + 3·37-s + 6·39-s − 7·41-s + 9·43-s + 6·45-s + 16·47-s + 7·49-s + 21·51-s − 12·53-s − 6·55-s − 3·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s − 1.13·7-s + 9-s − 0.904·11-s + 0.554·13-s + 1.54·15-s + 1.69·17-s − 0.229·19-s − 1.96·21-s + 1.45·23-s + 3/5·25-s + 0.928·29-s − 1.43·31-s − 1.56·33-s − 1.01·35-s + 0.493·37-s + 0.960·39-s − 1.09·41-s + 1.37·43-s + 0.894·45-s + 2.33·47-s + 49-s + 2.94·51-s − 1.64·53-s − 0.809·55-s − 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(67600\)    =    \(2^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.31023\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 67600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623599650\)
\(L(\frac12)\) \(\approx\) \(2.623599650\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
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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

−12.19853878755425947420649886630, −12.17554128371401982522629540324, −10.92717939296625100326435103603, −10.73318432269809254452033174415, −10.18152503572189814322031029344, −9.741117966510711587374384476735, −9.245426829227925592022661067875, −8.827687307576528227955970364243, −8.668143891591581185605195776330, −7.70986097863580839143133130907, −7.59608727551051792063104533307, −6.88566194168091777047892779835, −6.15033916662252742637679164847, −5.68511246731587629557269549968, −5.16667074973538874784221686285, −4.15126644541010437766477122150, −3.38456394538072386425508820427, −2.82742163210141865177049829200, −2.68229165966164450814091997592, −1.38598263478942632454871572922, 1.38598263478942632454871572922, 2.68229165966164450814091997592, 2.82742163210141865177049829200, 3.38456394538072386425508820427, 4.15126644541010437766477122150, 5.16667074973538874784221686285, 5.68511246731587629557269549968, 6.15033916662252742637679164847, 6.88566194168091777047892779835, 7.59608727551051792063104533307, 7.70986097863580839143133130907, 8.668143891591581185605195776330, 8.827687307576528227955970364243, 9.245426829227925592022661067875, 9.741117966510711587374384476735, 10.18152503572189814322031029344, 10.73318432269809254452033174415, 10.92717939296625100326435103603, 12.17554128371401982522629540324, 12.19853878755425947420649886630

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