Properties

Label 2-5e4-1.1-c1-0-17
Degree $2$
Conductor $625$
Sign $-1$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 3-s − 1.61·4-s + 0.618·6-s + 1.61·7-s + 2.23·8-s − 2·9-s − 0.763·11-s + 1.61·12-s + 4.85·13-s − 1.00·14-s + 1.85·16-s + 0.763·17-s + 1.23·18-s − 5.85·19-s − 1.61·21-s + 0.472·22-s − 8.23·23-s − 2.23·24-s − 3.00·26-s + 5·27-s − 2.61·28-s − 1.38·29-s − 3·31-s − 5.61·32-s + 0.763·33-s − 0.472·34-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.252·6-s + 0.611·7-s + 0.790·8-s − 0.666·9-s − 0.230·11-s + 0.467·12-s + 1.34·13-s − 0.267·14-s + 0.463·16-s + 0.185·17-s + 0.291·18-s − 1.34·19-s − 0.353·21-s + 0.100·22-s − 1.71·23-s − 0.456·24-s − 0.588·26-s + 0.962·27-s − 0.494·28-s − 0.256·29-s − 0.538·31-s − 0.993·32-s + 0.132·33-s − 0.0809·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-1$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 0.763T + 11T^{2} \)
13 \( 1 - 4.85T + 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 + 5.47T + 53T^{2} \)
59 \( 1 + 4.14T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 1.76T + 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
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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

−10.37836332430107856791355928771, −9.158769433767514207650242644238, −8.402407948670096481927558824071, −7.911494512284177911245429121248, −6.39846032466690188750117531412, −5.61936346762485966861314235998, −4.63158370589134938107945374811, −3.61982368548635112315187805330, −1.70627417996651129056379359719, 0, 1.70627417996651129056379359719, 3.61982368548635112315187805330, 4.63158370589134938107945374811, 5.61936346762485966861314235998, 6.39846032466690188750117531412, 7.911494512284177911245429121248, 8.402407948670096481927558824071, 9.158769433767514207650242644238, 10.37836332430107856791355928771

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