Properties

Label 4-475e2-1.1-c2e2-0-1
Degree $4$
Conductor $225625$
Sign $1$
Analytic cond. $167.516$
Root an. cond. $3.59761$
Motivic weight $2$
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  − 8·4-s − 18·9-s + 6·11-s + 48·16-s + 38·19-s + 144·36-s − 48·44-s + 73·49-s + 206·61-s − 256·64-s − 304·76-s + 243·81-s − 108·99-s − 204·101-s − 215·121-s + 127-s + 131-s + 137-s + 139-s − 864·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s − 684·171-s + ⋯
L(s)  = 1  − 2·4-s − 2·9-s + 6/11·11-s + 3·16-s + 2·19-s + 4·36-s − 1.09·44-s + 1.48·49-s + 3.37·61-s − 4·64-s − 4·76-s + 3·81-s − 1.09·99-s − 2.01·101-s − 1.77·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s − 4·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(225625\)    =    \(5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(167.516\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 225625,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9439256316\)
\(L(\frac12)\) \(\approx\) \(0.9439256316\)
\(L(2)\) 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 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good2$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 73 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 353 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 158 T^{2} + p^{4} T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 + 3527 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 1207 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 103 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 10033 T^{2} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2^2$ \( 1 - 5678 T^{2} + p^{4} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
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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

−10.96348772132453140814658003016, −10.55417973163687070222589555231, −9.898551259455338163206544257748, −9.550768301795540069324048842231, −9.237904884078837185636814820836, −8.832656450362261840594529375692, −8.260771632350344332623621372625, −8.249656551299161358940077699160, −7.52347679468063298774791178230, −6.94912980040000317484386554876, −6.12877410451595404205008172589, −5.64315199029482536239989573967, −5.25321551886673374653250111097, −5.08250906432001103913579137753, −4.11342820987329293886349479990, −3.74022800459851719505078817882, −3.18221161227009083577586528517, −2.54783407496668986615495354907, −1.13332464440955089542180359985, −0.49320274841841272145180973859, 0.49320274841841272145180973859, 1.13332464440955089542180359985, 2.54783407496668986615495354907, 3.18221161227009083577586528517, 3.74022800459851719505078817882, 4.11342820987329293886349479990, 5.08250906432001103913579137753, 5.25321551886673374653250111097, 5.64315199029482536239989573967, 6.12877410451595404205008172589, 6.94912980040000317484386554876, 7.52347679468063298774791178230, 8.249656551299161358940077699160, 8.260771632350344332623621372625, 8.832656450362261840594529375692, 9.237904884078837185636814820836, 9.550768301795540069324048842231, 9.898551259455338163206544257748, 10.55417973163687070222589555231, 10.96348772132453140814658003016

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