Properties

Degree $4$
Conductor $3125$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11-s − 4·16-s − 11·31-s + 9·41-s − 61-s + 19·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 0.301·11-s − 16-s − 1.97·31-s + 1.40·41-s − 0.128·61-s + 2.25·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.301·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3125\)    =    \(5^{5}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3125} (1, \cdot )$
Sato-Tate group: $F_{ac}$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3125,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7183136284\)
\(L(\frac12)\) \(\approx\) \(0.7183136284\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_4$ \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_4$ \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + p^{2} T^{4} \)
41$C_4$ \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_4$ \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_4$ \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + p^{2} T^{4} \)
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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

−18.2513478698, −17.7940303180, −17.0572865553, −16.6516421521, −15.9944202971, −15.6665478707, −15.0412242767, −14.3272056793, −14.0663480476, −13.2082005437, −12.8575532333, −12.2756492446, −11.4301570976, −11.0758025400, −10.4664572499, −9.62470469199, −9.14526688174, −8.48931188846, −7.62285385181, −7.10218589390, −6.22989238233, −5.43292897511, −4.57292147662, −3.58578684104, −2.26074119601, 2.26074119601, 3.58578684104, 4.57292147662, 5.43292897511, 6.22989238233, 7.10218589390, 7.62285385181, 8.48931188846, 9.14526688174, 9.62470469199, 10.4664572499, 11.0758025400, 11.4301570976, 12.2756492446, 12.8575532333, 13.2082005437, 14.0663480476, 14.3272056793, 15.0412242767, 15.6665478707, 15.9944202971, 16.6516421521, 17.0572865553, 17.7940303180, 18.2513478698

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