Properties

Label 4-5070e2-1.1-c1e2-0-5
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 12·17-s − 16·23-s − 25-s − 4·27-s + 12·29-s − 3·36-s − 8·43-s − 2·48-s + 14·49-s − 24·51-s − 20·53-s − 4·61-s − 64-s − 12·68-s + 32·69-s + 2·75-s − 32·79-s + 5·81-s − 24·87-s + 16·92-s + 100-s + 4·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 2.91·17-s − 3.33·23-s − 1/5·25-s − 0.769·27-s + 2.22·29-s − 1/2·36-s − 1.21·43-s − 0.288·48-s + 2·49-s − 3.36·51-s − 2.74·53-s − 0.512·61-s − 1/8·64-s − 1.45·68-s + 3.85·69-s + 0.230·75-s − 3.60·79-s + 5/9·81-s − 2.57·87-s + 1.66·92-s + 1/10·100-s + 0.398·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9651385217\)
\(L(\frac12)\) \(\approx\) \(0.9651385217\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.322744553475654579019178608893, −8.072109998916235197105837615063, −7.63143783172067378376054679059, −7.50233812875175255108011869558, −6.89345169608282922067128485984, −6.53871494493932803496032202455, −5.97208297641197493495612767500, −5.90255786825983424374140417080, −5.67281168118077168719861733110, −5.21464337660547116155464114702, −4.60823928175496984421791453168, −4.56725924546548230116991366418, −3.90967077195863446539368182483, −3.73093990139767201189774484145, −3.01022719892157244847149251919, −2.86520048737423861042903124174, −1.85225113094165631182200785856, −1.56538039233603013895554301316, −0.985622654308429601258639247182, −0.34515317747471623257156338591, 0.34515317747471623257156338591, 0.985622654308429601258639247182, 1.56538039233603013895554301316, 1.85225113094165631182200785856, 2.86520048737423861042903124174, 3.01022719892157244847149251919, 3.73093990139767201189774484145, 3.90967077195863446539368182483, 4.56725924546548230116991366418, 4.60823928175496984421791453168, 5.21464337660547116155464114702, 5.67281168118077168719861733110, 5.90255786825983424374140417080, 5.97208297641197493495612767500, 6.53871494493932803496032202455, 6.89345169608282922067128485984, 7.50233812875175255108011869558, 7.63143783172067378376054679059, 8.072109998916235197105837615063, 8.322744553475654579019178608893

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