Properties

Label 4-75e2-1.1-c1e2-0-0
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  − 9-s + 4·11-s − 4·16-s + 10·19-s − 20·29-s − 6·31-s − 16·41-s + 5·49-s + 20·59-s + 14·61-s − 16·71-s + 81-s − 4·99-s + 24·101-s − 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.20·11-s − 16-s + 2.29·19-s − 3.71·29-s − 1.07·31-s − 2.49·41-s + 5/7·49-s + 2.60·59-s + 1.79·61-s − 1.89·71-s + 1/9·81-s − 0.402·99-s + 2.38·101-s − 0.957·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-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

\(L(1)\) \(\approx\) \(0.8721814436\)
\(L(\frac12)\) \(\approx\) \(0.8721814436\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−14.91080771710218934571399627491, −14.26732083822925441082468851828, −13.76560097149990409305510756326, −13.21515765633412552018180043720, −12.82684476259162027079655811408, −11.74278915826813398513090580853, −11.47012906506001366604380880864, −11.44560299852911851411715448857, −10.28050345487040540000848756328, −9.715877541712923973372115428251, −8.998851506099230354236081802260, −8.942075476039868462130469896912, −7.75232414789649101971239372127, −7.19132413134035066790980104844, −6.73196839614468195283312946179, −5.50602029648065362711011946692, −5.36869590813685455208628735030, −3.95467184940752398421650199006, −3.42992584500941121446531627913, −1.87406895516215122089339214148, 1.87406895516215122089339214148, 3.42992584500941121446531627913, 3.95467184940752398421650199006, 5.36869590813685455208628735030, 5.50602029648065362711011946692, 6.73196839614468195283312946179, 7.19132413134035066790980104844, 7.75232414789649101971239372127, 8.942075476039868462130469896912, 8.998851506099230354236081802260, 9.715877541712923973372115428251, 10.28050345487040540000848756328, 11.44560299852911851411715448857, 11.47012906506001366604380880864, 11.74278915826813398513090580853, 12.82684476259162027079655811408, 13.21515765633412552018180043720, 13.76560097149990409305510756326, 14.26732083822925441082468851828, 14.91080771710218934571399627491

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