Properties

Label 2-75e2-1.1-c1-0-153
Degree $2$
Conductor $5625$
Sign $-1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
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  − 0.856·2-s − 1.26·4-s + 0.647·7-s + 2.79·8-s + 5.91·11-s + 2.88·13-s − 0.555·14-s + 0.134·16-s − 4.20·17-s − 1.64·19-s − 5.06·22-s − 6.42·23-s − 2.47·26-s − 0.820·28-s − 2.25·29-s − 5.28·31-s − 5.71·32-s + 3.59·34-s − 1.71·37-s + 1.41·38-s + 0.176·41-s + 7.95·43-s − 7.48·44-s + 5.50·46-s − 1.05·47-s − 6.58·49-s − 3.65·52-s + ⋯
L(s)  = 1  − 0.605·2-s − 0.632·4-s + 0.244·7-s + 0.989·8-s + 1.78·11-s + 0.799·13-s − 0.148·14-s + 0.0336·16-s − 1.01·17-s − 0.378·19-s − 1.08·22-s − 1.34·23-s − 0.484·26-s − 0.155·28-s − 0.419·29-s − 0.949·31-s − 1.00·32-s + 0.617·34-s − 0.282·37-s + 0.229·38-s + 0.0275·41-s + 1.21·43-s − 1.12·44-s + 0.811·46-s − 0.154·47-s − 0.940·49-s − 0.506·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5625} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 0.856T + 2T^{2} \)
7 \( 1 - 0.647T + 7T^{2} \)
11 \( 1 - 5.91T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + 1.64T + 19T^{2} \)
23 \( 1 + 6.42T + 23T^{2} \)
29 \( 1 + 2.25T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + 1.71T + 37T^{2} \)
41 \( 1 - 0.176T + 41T^{2} \)
43 \( 1 - 7.95T + 43T^{2} \)
47 \( 1 + 1.05T + 47T^{2} \)
53 \( 1 - 4.48T + 53T^{2} \)
59 \( 1 + 9.74T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 6.09T + 71T^{2} \)
73 \( 1 + 7.08T + 73T^{2} \)
79 \( 1 - 1.58T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 7.55T + 97T^{2} \)
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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

−7.915329046153441605737951049442, −7.20839637272848105872288997618, −6.33287821624791713122768357976, −5.82326375412951021396540915415, −4.60436356089788232134304599221, −4.14708626694858821789731370357, −3.49125378666243292548000194798, −1.94151306311571515090739530360, −1.31384229179971658256029510171, 0, 1.31384229179971658256029510171, 1.94151306311571515090739530360, 3.49125378666243292548000194798, 4.14708626694858821789731370357, 4.60436356089788232134304599221, 5.82326375412951021396540915415, 6.33287821624791713122768357976, 7.20839637272848105872288997618, 7.915329046153441605737951049442

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