Properties

Label 4-1260e2-1.1-c1e2-0-38
Degree $4$
Conductor $1587600$
Sign $1$
Analytic cond. $101.226$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 2·11-s − 12·13-s + 2·17-s − 9·23-s − 6·29-s − 2·31-s − 35-s − 8·37-s − 10·41-s + 2·43-s + 8·47-s − 6·49-s + 4·53-s + 2·55-s − 8·59-s − 7·61-s + 12·65-s + 3·67-s − 16·71-s − 14·73-s − 2·77-s − 4·79-s + 2·83-s − 2·85-s + 13·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.603·11-s − 3.32·13-s + 0.485·17-s − 1.87·23-s − 1.11·29-s − 0.359·31-s − 0.169·35-s − 1.31·37-s − 1.56·41-s + 0.304·43-s + 1.16·47-s − 6/7·49-s + 0.549·53-s + 0.269·55-s − 1.04·59-s − 0.896·61-s + 1.48·65-s + 0.366·67-s − 1.89·71-s − 1.63·73-s − 0.227·77-s − 0.450·79-s + 0.219·83-s − 0.216·85-s + 1.37·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1587600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(101.226\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1587600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.461088331473120875808656168144, −9.309868891136662265744523179206, −8.511392212267655659554470876859, −8.243255835849661930905107090134, −7.69402221458071211565812372635, −7.53815653818129945143118025082, −7.05501691172677050728245760253, −6.95589143197288856539100214003, −5.87907712310243462831817296094, −5.77308167799529509719522802573, −5.02780948873472597040228352468, −4.98181939404984270510884788997, −4.29768152399406475569215937741, −3.96442610923289746335142871840, −3.10052016296613180675074930278, −2.79670213459983831028136736420, −2.00867417997297657111575291231, −1.75206441701054846215614564285, 0, 0, 1.75206441701054846215614564285, 2.00867417997297657111575291231, 2.79670213459983831028136736420, 3.10052016296613180675074930278, 3.96442610923289746335142871840, 4.29768152399406475569215937741, 4.98181939404984270510884788997, 5.02780948873472597040228352468, 5.77308167799529509719522802573, 5.87907712310243462831817296094, 6.95589143197288856539100214003, 7.05501691172677050728245760253, 7.53815653818129945143118025082, 7.69402221458071211565812372635, 8.243255835849661930905107090134, 8.511392212267655659554470876859, 9.309868891136662265744523179206, 9.461088331473120875808656168144

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