Properties

Label 2-65e2-1.1-c1-0-58
Degree $2$
Conductor $4225$
Sign $1$
Analytic cond. $33.7367$
Root an. cond. $5.80833$
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  + 0.219·2-s + 1.60·3-s − 1.95·4-s + 0.351·6-s + 0.332·7-s − 0.868·8-s − 0.439·9-s − 5.37·11-s − 3.12·12-s + 0.0729·14-s + 3.71·16-s + 5.06·17-s − 0.0965·18-s + 2.26·19-s + 0.531·21-s − 1.18·22-s + 2.83·23-s − 1.38·24-s − 5.50·27-s − 0.648·28-s − 2.90·29-s + 5.46·31-s + 2.55·32-s − 8.59·33-s + 1.11·34-s + 0.857·36-s − 5.97·37-s + ⋯
L(s)  = 1  + 0.155·2-s + 0.923·3-s − 0.975·4-s + 0.143·6-s + 0.125·7-s − 0.306·8-s − 0.146·9-s − 1.61·11-s − 0.901·12-s + 0.0195·14-s + 0.928·16-s + 1.22·17-s − 0.0227·18-s + 0.520·19-s + 0.116·21-s − 0.251·22-s + 0.592·23-s − 0.283·24-s − 1.05·27-s − 0.122·28-s − 0.539·29-s + 0.981·31-s + 0.451·32-s − 1.49·33-s + 0.190·34-s + 0.142·36-s − 0.981·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(33.7367\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.834558568\)
\(L(\frac12)\) \(\approx\) \(1.834558568\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 - 0.219T + 2T^{2} \)
3 \( 1 - 1.60T + 3T^{2} \)
7 \( 1 - 0.332T + 7T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 5.97T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 5.06T + 43T^{2} \)
47 \( 1 + 8.34T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 9.68T + 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 - 4.26T + 83T^{2} \)
89 \( 1 - 3.22T + 89T^{2} \)
97 \( 1 - 2.50T + 97T^{2} \)
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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

−8.215598351319461238884186749965, −8.002313615481687514612614036374, −7.25500620083369787390675465520, −5.93954391346405088334786803038, −5.28009847982693619413447572296, −4.76909852104274917697264579268, −3.53428883437294555433550297231, −3.15550906216449768702011007655, −2.18541160804132786755557800103, −0.70791127681211863382592782506, 0.70791127681211863382592782506, 2.18541160804132786755557800103, 3.15550906216449768702011007655, 3.53428883437294555433550297231, 4.76909852104274917697264579268, 5.28009847982693619413447572296, 5.93954391346405088334786803038, 7.25500620083369787390675465520, 8.002313615481687514612614036374, 8.215598351319461238884186749965

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