Properties

Label 2-75e2-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.16·2-s + 2.66·4-s − 3.16·7-s − 1.44·8-s − 1.53·11-s − 5.24·13-s + 6.82·14-s − 2.21·16-s − 1.29·17-s + 5.44·19-s + 3.32·22-s + 6.44·23-s + 11.3·26-s − 8.43·28-s + 2.36·29-s − 4.46·31-s + 7.67·32-s + 2.78·34-s − 5.95·37-s − 11.7·38-s + 8.53·41-s − 8.48·43-s − 4.10·44-s − 13.9·46-s − 0.753·47-s + 2.98·49-s − 13.9·52-s + ⋯
L(s)  = 1  − 1.52·2-s + 1.33·4-s − 1.19·7-s − 0.510·8-s − 0.463·11-s − 1.45·13-s + 1.82·14-s − 0.554·16-s − 0.313·17-s + 1.24·19-s + 0.708·22-s + 1.34·23-s + 2.22·26-s − 1.59·28-s + 0.438·29-s − 0.802·31-s + 1.35·32-s + 0.478·34-s − 0.979·37-s − 1.90·38-s + 1.33·41-s − 1.29·43-s − 0.618·44-s − 2.05·46-s − 0.109·47-s + 0.427·49-s − 1.93·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3450730032\)
\(L(\frac12)\) \(\approx\) \(0.3450730032\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.16T + 2T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 1.53T + 11T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
23 \( 1 - 6.44T + 23T^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
31 \( 1 + 4.46T + 31T^{2} \)
37 \( 1 + 5.95T + 37T^{2} \)
41 \( 1 - 8.53T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 0.753T + 47T^{2} \)
53 \( 1 + 9.74T + 53T^{2} \)
59 \( 1 + 4.11T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 1.89T + 67T^{2} \)
71 \( 1 - 0.0708T + 71T^{2} \)
73 \( 1 - 4.01T + 73T^{2} \)
79 \( 1 - 1.61T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + 7.27T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.182723767880632420995213679492, −7.37286883526172615101535209726, −7.11710655025263157753633962579, −6.34515510178724976371367562194, −5.30480067591259279914625458998, −4.61182005503205383424322945826, −3.21610097849023446568333709148, −2.72613144922462233325021435196, −1.60324013865665778616347200561, −0.39120460119267850683692091073, 0.39120460119267850683692091073, 1.60324013865665778616347200561, 2.72613144922462233325021435196, 3.21610097849023446568333709148, 4.61182005503205383424322945826, 5.30480067591259279914625458998, 6.34515510178724976371367562194, 7.11710655025263157753633962579, 7.37286883526172615101535209726, 8.182723767880632420995213679492

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