Properties

Label 4-2475-1.1-c1e2-0-0
Degree $4$
Conductor $2475$
Sign $1$
Analytic cond. $0.157808$
Root an. cond. $0.630278$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·5-s + 9-s − 3·11-s − 3·16-s − 2·20-s − 25-s + 4·29-s + 36-s − 4·41-s − 3·44-s − 2·45-s + 2·49-s + 6·55-s + 24·59-s − 4·61-s − 7·64-s + 24·79-s + 6·80-s + 81-s − 12·89-s − 3·99-s − 100-s − 12·101-s − 20·109-s + 4·116-s + 2·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 1/3·9-s − 0.904·11-s − 3/4·16-s − 0.447·20-s − 1/5·25-s + 0.742·29-s + 1/6·36-s − 0.624·41-s − 0.452·44-s − 0.298·45-s + 2/7·49-s + 0.809·55-s + 3.12·59-s − 0.512·61-s − 7/8·64-s + 2.70·79-s + 0.670·80-s + 1/9·81-s − 1.27·89-s − 0.301·99-s − 0.0999·100-s − 1.19·101-s − 1.91·109-s + 0.371·116-s + 2/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.157808\)
Root analytic conductor: \(0.630278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2475,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6606441068\)
\(L(\frac12)\) \(\approx\) \(0.6606441068\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
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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

−13.14575241523544191772257609292, −12.23662699462035013540508107849, −11.93851842213943004021314056240, −11.22694089429559050080949866662, −10.73683235986961041263444060107, −10.08219127334790378188359306081, −9.355764873253690620092615692808, −8.396627232303075215483887640772, −7.987064380804719908863906390256, −7.14294632569963089923076678235, −6.67514988045421682235703457049, −5.55778831301744703808824925181, −4.66431952886322862521445388232, −3.71584900926351816916902784454, −2.45765074847302232037845071651, 2.45765074847302232037845071651, 3.71584900926351816916902784454, 4.66431952886322862521445388232, 5.55778831301744703808824925181, 6.67514988045421682235703457049, 7.14294632569963089923076678235, 7.987064380804719908863906390256, 8.396627232303075215483887640772, 9.355764873253690620092615692808, 10.08219127334790378188359306081, 10.73683235986961041263444060107, 11.22694089429559050080949866662, 11.93851842213943004021314056240, 12.23662699462035013540508107849, 13.14575241523544191772257609292

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