Properties

Label 4-2925e2-1.1-c1e2-0-9
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $545.514$
Root an. cond. $4.83282$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·11-s − 4·16-s − 4·19-s + 20·29-s − 4·31-s + 18·41-s + 5·49-s − 22·61-s − 30·71-s + 22·79-s − 22·89-s + 24·101-s − 32·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 40·176-s + 179-s + ⋯
L(s)  = 1  + 3.01·11-s − 16-s − 0.917·19-s + 3.71·29-s − 0.718·31-s + 2.81·41-s + 5/7·49-s − 2.81·61-s − 3.56·71-s + 2.47·79-s − 2.33·89-s + 2.38·101-s − 3.06·109-s + 4.81·121-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 − 0.0769·169-s + 0.0760·173-s − 3.01·176-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(545.514\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.278508620\)
\(L(\frac12)\) \(\approx\) \(3.278508620\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
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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

−8.884456166963231014604933930854, −8.675566139624296525773197682312, −8.452786135679513960131907016705, −7.66492130684733042809321401829, −7.49686593032851406998557892223, −6.88118108496805208111348224235, −6.66144753550010207973464178678, −6.27135319440274831991477037726, −6.12942240015596643193643312437, −5.71290059337612206925382099007, −4.69536624586676876610615779475, −4.67030000607447335102972201670, −4.22748431481126597095443123280, −3.99007128453281432375083029977, −3.40357829694951989747006121101, −2.66787646879889425397448257677, −2.56204718516906521627184556147, −1.53472755317651973535059309969, −1.34851762432787253956217612460, −0.62469013096305254382519254061, 0.62469013096305254382519254061, 1.34851762432787253956217612460, 1.53472755317651973535059309969, 2.56204718516906521627184556147, 2.66787646879889425397448257677, 3.40357829694951989747006121101, 3.99007128453281432375083029977, 4.22748431481126597095443123280, 4.67030000607447335102972201670, 4.69536624586676876610615779475, 5.71290059337612206925382099007, 6.12942240015596643193643312437, 6.27135319440274831991477037726, 6.66144753550010207973464178678, 6.88118108496805208111348224235, 7.49686593032851406998557892223, 7.66492130684733042809321401829, 8.452786135679513960131907016705, 8.675566139624296525773197682312, 8.884456166963231014604933930854

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