Properties

Label 2-65e2-1.1-c1-0-25
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  + 1.49·2-s − 0.0947·3-s + 0.236·4-s − 0.141·6-s − 4.82·7-s − 2.63·8-s − 2.99·9-s − 1.06·11-s − 0.0224·12-s − 7.21·14-s − 4.41·16-s − 3.55·17-s − 4.47·18-s + 5.73·19-s + 0.457·21-s − 1.59·22-s + 7.08·23-s + 0.249·24-s + 0.567·27-s − 1.14·28-s + 1.47·29-s − 1.46·31-s − 1.33·32-s + 0.101·33-s − 5.32·34-s − 0.707·36-s + 0.0253·37-s + ⋯
L(s)  = 1  + 1.05·2-s − 0.0547·3-s + 0.118·4-s − 0.0578·6-s − 1.82·7-s − 0.932·8-s − 0.997·9-s − 0.322·11-s − 0.00647·12-s − 1.92·14-s − 1.10·16-s − 0.863·17-s − 1.05·18-s + 1.31·19-s + 0.0998·21-s − 0.340·22-s + 1.47·23-s + 0.0510·24-s + 0.109·27-s − 0.215·28-s + 0.273·29-s − 0.262·31-s − 0.235·32-s + 0.0176·33-s − 0.912·34-s − 0.117·36-s + 0.00417·37-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

\(L(1)\) \(\approx\) \(1.309576634\)
\(L(\frac12)\) \(\approx\) \(1.309576634\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.49T + 2T^{2} \)
3 \( 1 + 0.0947T + 3T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
11 \( 1 + 1.06T + 11T^{2} \)
17 \( 1 + 3.55T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 - 7.08T + 23T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 0.0253T + 37T^{2} \)
41 \( 1 - 0.267T + 41T^{2} \)
43 \( 1 + 3.55T + 43T^{2} \)
47 \( 1 + 6.51T + 47T^{2} \)
53 \( 1 + 0.991T + 53T^{2} \)
59 \( 1 - 8.72T + 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
67 \( 1 + 5.17T + 67T^{2} \)
71 \( 1 + 7.76T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 - 0.725T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 3.43T + 97T^{2} \)
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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

−8.650261659655597102594982663960, −7.39241028749774297801459400728, −6.64621538943561153430898291545, −6.11121323628171498725595707375, −5.38464026816075382585926093840, −4.77289239589637991497589320890, −3.60075717165485199145453344471, −3.16091299860889454942666357025, −2.55044516287328545730059277893, −0.51913145495065955529476892799, 0.51913145495065955529476892799, 2.55044516287328545730059277893, 3.16091299860889454942666357025, 3.60075717165485199145453344471, 4.77289239589637991497589320890, 5.38464026816075382585926093840, 6.11121323628171498725595707375, 6.64621538943561153430898291545, 7.39241028749774297801459400728, 8.650261659655597102594982663960

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