Properties

Label 4-75e2-1.1-c1e2-0-2
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $0.358654$
Root an. cond. $0.773872$
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  + 3·4-s − 9-s − 8·11-s + 5·16-s − 8·19-s + 4·29-s − 3·36-s + 20·41-s − 24·44-s + 14·49-s + 8·59-s − 4·61-s + 3·64-s − 16·71-s − 24·76-s + 81-s + 12·89-s + 8·99-s + 12·101-s − 28·109-s + 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯
L(s)  = 1  + 3/2·4-s − 1/3·9-s − 2.41·11-s + 5/4·16-s − 1.83·19-s + 0.742·29-s − 1/2·36-s + 3.12·41-s − 3.61·44-s + 2·49-s + 1.04·59-s − 0.512·61-s + 3/8·64-s − 1.89·71-s − 2.75·76-s + 1/9·81-s + 1.27·89-s + 0.804·99-s + 1.19·101-s − 2.68·109-s + 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.358654\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{75} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019195692\)
\(L(\frac12)\) \(\approx\) \(1.019195692\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−14.82600110360146444091838197656, −14.56809004589719542184363179136, −13.60501366116669137684821098129, −13.12990429596880409303772465604, −12.59166450815821389290892738729, −12.19251527247341498052026940081, −11.37619323081471910438366922608, −10.91103772036436981090062822670, −10.40433512253293882895387196372, −10.30336279752583334125713240413, −9.066268324289895262971809662894, −8.436983745411476596171759227745, −7.59119348029160234777087323207, −7.52156633457050854866829125129, −6.43506067565300041466428256615, −5.95632579760025348206124952315, −5.21819757671916085151264765739, −4.17314498147057539715128367537, −2.59227805750375734226524276590, −2.50577656933738056614132106187, 2.50577656933738056614132106187, 2.59227805750375734226524276590, 4.17314498147057539715128367537, 5.21819757671916085151264765739, 5.95632579760025348206124952315, 6.43506067565300041466428256615, 7.52156633457050854866829125129, 7.59119348029160234777087323207, 8.436983745411476596171759227745, 9.066268324289895262971809662894, 10.30336279752583334125713240413, 10.40433512253293882895387196372, 10.91103772036436981090062822670, 11.37619323081471910438366922608, 12.19251527247341498052026940081, 12.59166450815821389290892738729, 13.12990429596880409303772465604, 13.60501366116669137684821098129, 14.56809004589719542184363179136, 14.82600110360146444091838197656

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