Properties

Label 2-5e4-1.1-c7-0-79
Degree $2$
Conductor $625$
Sign $-1$
Analytic cond. $195.240$
Root an. cond. $13.9728$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.3·2-s − 85.0·3-s + 173.·4-s + 1.47e3·6-s − 1.27e3·7-s − 794.·8-s + 5.04e3·9-s − 4.57e3·11-s − 1.47e4·12-s − 2.91e3·13-s + 2.21e4·14-s − 8.44e3·16-s + 1.44e4·17-s − 8.76e4·18-s + 4.44e4·19-s + 1.08e5·21-s + 7.95e4·22-s − 9.81e4·23-s + 6.75e4·24-s + 5.05e4·26-s − 2.43e5·27-s − 2.21e5·28-s − 1.01e5·29-s + 1.08e5·31-s + 2.48e5·32-s + 3.89e5·33-s − 2.51e5·34-s + ⋯
L(s)  = 1  − 1.53·2-s − 1.81·3-s + 1.35·4-s + 2.79·6-s − 1.40·7-s − 0.548·8-s + 2.30·9-s − 1.03·11-s − 2.46·12-s − 0.367·13-s + 2.16·14-s − 0.515·16-s + 0.713·17-s − 3.54·18-s + 1.48·19-s + 2.56·21-s + 1.59·22-s − 1.68·23-s + 0.997·24-s + 0.564·26-s − 2.37·27-s − 1.91·28-s − 0.773·29-s + 0.654·31-s + 1.33·32-s + 1.88·33-s − 1.09·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-1$
Analytic conductor: \(195.240\)
Root analytic conductor: \(13.9728\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 625,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 17.3T + 128T^{2} \)
3 \( 1 + 85.0T + 2.18e3T^{2} \)
7 \( 1 + 1.27e3T + 8.23e5T^{2} \)
11 \( 1 + 4.57e3T + 1.94e7T^{2} \)
13 \( 1 + 2.91e3T + 6.27e7T^{2} \)
17 \( 1 - 1.44e4T + 4.10e8T^{2} \)
19 \( 1 - 4.44e4T + 8.93e8T^{2} \)
23 \( 1 + 9.81e4T + 3.40e9T^{2} \)
29 \( 1 + 1.01e5T + 1.72e10T^{2} \)
31 \( 1 - 1.08e5T + 2.75e10T^{2} \)
37 \( 1 - 3.50e5T + 9.49e10T^{2} \)
41 \( 1 + 5.35e5T + 1.94e11T^{2} \)
43 \( 1 + 7.42e5T + 2.71e11T^{2} \)
47 \( 1 + 7.88e5T + 5.06e11T^{2} \)
53 \( 1 + 9.56e5T + 1.17e12T^{2} \)
59 \( 1 - 4.81e5T + 2.48e12T^{2} \)
61 \( 1 + 2.21e6T + 3.14e12T^{2} \)
67 \( 1 + 3.01e6T + 6.06e12T^{2} \)
71 \( 1 - 1.42e6T + 9.09e12T^{2} \)
73 \( 1 - 4.21e6T + 1.10e13T^{2} \)
79 \( 1 + 4.47e6T + 1.92e13T^{2} \)
83 \( 1 + 1.18e6T + 2.71e13T^{2} \)
89 \( 1 - 3.18e6T + 4.42e13T^{2} \)
97 \( 1 + 4.00e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.641781149548704699442439440529, −8.013657308478750179034841152765, −7.34205524474357452274950210180, −6.46639691109541079539838591006, −5.76438777786270388267393062290, −4.78158547765499136044629697103, −3.22082282096043021282403402296, −1.70307678129946264531724824047, −0.57440163279085088200464474643, 0, 0.57440163279085088200464474643, 1.70307678129946264531724824047, 3.22082282096043021282403402296, 4.78158547765499136044629697103, 5.76438777786270388267393062290, 6.46639691109541079539838591006, 7.34205524474357452274950210180, 8.013657308478750179034841152765, 9.641781149548704699442439440529

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