Properties

Label 4-560e2-1.1-c1e2-0-50
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
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  − 4·5-s − 3·9-s − 6·11-s − 16·19-s + 11·25-s + 2·29-s + 4·31-s − 12·41-s + 12·45-s − 49-s + 24·55-s − 20·59-s − 14·79-s − 16·89-s + 64·95-s + 18·99-s − 24·101-s + 14·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s − 1.80·11-s − 3.67·19-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 1.87·41-s + 1.78·45-s − 1/7·49-s + 3.23·55-s − 2.60·59-s − 1.57·79-s − 1.69·89-s + 6.56·95-s + 1.80·99-s − 2.38·101-s + 1.34·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 185 T^{2} + p^{2} T^{4} \)
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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

−10.58804003912029222761714093689, −10.39122221674609264460421941462, −9.813340107754439583591978097041, −8.835539964443643956614724393747, −8.673555457906657883500618249836, −8.199690345447704683467608904797, −8.129436152953682363906960379319, −7.58129235973462725917552446622, −6.90625967382598233767942439212, −6.54283079933294675299992408879, −6.00644602808738444314514610434, −5.36047721508849218345547444582, −4.72136855038995652583907868073, −4.40118322555487914118174305000, −3.91813131858336118609679447058, −2.90182251378309740686117137120, −2.89313021270953584683833638981, −1.90029477991754274331086169687, 0, 0, 1.90029477991754274331086169687, 2.89313021270953584683833638981, 2.90182251378309740686117137120, 3.91813131858336118609679447058, 4.40118322555487914118174305000, 4.72136855038995652583907868073, 5.36047721508849218345547444582, 6.00644602808738444314514610434, 6.54283079933294675299992408879, 6.90625967382598233767942439212, 7.58129235973462725917552446622, 8.129436152953682363906960379319, 8.199690345447704683467608904797, 8.673555457906657883500618249836, 8.835539964443643956614724393747, 9.813340107754439583591978097041, 10.39122221674609264460421941462, 10.58804003912029222761714093689

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