Properties

Label 4-575e2-1.1-c1e2-0-2
Degree $4$
Conductor $330625$
Sign $1$
Analytic cond. $21.0809$
Root an. cond. $2.14275$
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  + 3·2-s + 2·3-s + 4·4-s + 6·6-s + 2·7-s + 3·8-s − 3·9-s − 2·11-s + 8·12-s + 8·13-s + 6·14-s + 3·16-s + 4·17-s − 9·18-s + 2·19-s + 4·21-s − 6·22-s + 2·23-s + 6·24-s + 24·26-s − 14·27-s + 8·28-s − 10·29-s + 4·31-s + 6·32-s − 4·33-s + 12·34-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 2·4-s + 2.44·6-s + 0.755·7-s + 1.06·8-s − 9-s − 0.603·11-s + 2.30·12-s + 2.21·13-s + 1.60·14-s + 3/4·16-s + 0.970·17-s − 2.12·18-s + 0.458·19-s + 0.872·21-s − 1.27·22-s + 0.417·23-s + 1.22·24-s + 4.70·26-s − 2.69·27-s + 1.51·28-s − 1.85·29-s + 0.718·31-s + 1.06·32-s − 0.696·33-s + 2.05·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(330625\)    =    \(5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(21.0809\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 330625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.542132585\)
\(L(\frac12)\) \(\approx\) \(8.542132585\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 22 T + 274 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−11.10911794872411536474810803491, −10.79665287608434845988667564892, −10.21328943056263325275742851703, −9.507468800920776256282332704371, −9.063656034039517945008518621852, −8.703618793056692228215169577434, −8.014013849197658166476626857420, −7.997255849139558219268896890065, −7.58595103679307296476772484357, −6.59145076816759939765121746332, −6.11036377884088353359100696984, −5.69954350468233113796506375934, −5.27294936805960046467214722930, −5.05762884202645613650587765067, −3.95679090222986520805569708975, −3.94719273499862893396810960549, −3.13516082241924280439628057420, −3.11800972478402242933589508765, −2.19820201054592301214911492646, −1.31465765825216604923597806682, 1.31465765825216604923597806682, 2.19820201054592301214911492646, 3.11800972478402242933589508765, 3.13516082241924280439628057420, 3.94719273499862893396810960549, 3.95679090222986520805569708975, 5.05762884202645613650587765067, 5.27294936805960046467214722930, 5.69954350468233113796506375934, 6.11036377884088353359100696984, 6.59145076816759939765121746332, 7.58595103679307296476772484357, 7.997255849139558219268896890065, 8.014013849197658166476626857420, 8.703618793056692228215169577434, 9.063656034039517945008518621852, 9.507468800920776256282332704371, 10.21328943056263325275742851703, 10.79665287608434845988667564892, 11.10911794872411536474810803491

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