Properties

Label 2-65e2-1.1-c1-0-12
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  − 2.51·2-s − 1.51·3-s + 4.32·4-s + 3.80·6-s − 3.32·7-s − 5.83·8-s − 0.707·9-s − 2.83·11-s − 6.54·12-s + 8.34·14-s + 6.02·16-s + 6.64·17-s + 1.77·18-s + 2.19·19-s + 5.02·21-s + 7.12·22-s + 0.485·23-s + 8.83·24-s + 5.61·27-s − 14.3·28-s − 3.32·29-s + 3.80·31-s − 3.48·32-s + 4.29·33-s − 16.6·34-s − 3.05·36-s − 9.32·37-s + ⋯
L(s)  = 1  − 1.77·2-s − 0.874·3-s + 2.16·4-s + 1.55·6-s − 1.25·7-s − 2.06·8-s − 0.235·9-s − 0.854·11-s − 1.88·12-s + 2.23·14-s + 1.50·16-s + 1.61·17-s + 0.419·18-s + 0.503·19-s + 1.09·21-s + 1.51·22-s + 0.101·23-s + 1.80·24-s + 1.08·27-s − 2.71·28-s − 0.616·29-s + 0.683·31-s − 0.616·32-s + 0.747·33-s − 2.86·34-s − 0.509·36-s − 1.53·37-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2054207245\)
\(L(\frac12)\) \(\approx\) \(0.2054207245\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 + 1.51T + 3T^{2} \)
7 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 + 2.83T + 11T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 - 2.19T + 19T^{2} \)
23 \( 1 - 0.485T + 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 - 3.80T + 31T^{2} \)
37 \( 1 + 9.32T + 37T^{2} \)
41 \( 1 + 1.61T + 41T^{2} \)
43 \( 1 - 0.872T + 43T^{2} \)
47 \( 1 - 3.32T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 8.83T + 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 4.29T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 0.585T + 79T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 + 0.641T + 97T^{2} \)
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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

−8.460248418746983545814230506125, −7.68171201806343560382694911251, −7.17398755556371871064532160267, −6.29150447097586289436863593571, −5.82954770205210196524129766837, −4.98944671111763772154955910844, −3.32345113219623643537240098311, −2.81789736381860156741099082775, −1.44957160891615253090413655208, −0.36190936849786380704763885174, 0.36190936849786380704763885174, 1.44957160891615253090413655208, 2.81789736381860156741099082775, 3.32345113219623643537240098311, 4.98944671111763772154955910844, 5.82954770205210196524129766837, 6.29150447097586289436863593571, 7.17398755556371871064532160267, 7.68171201806343560382694911251, 8.460248418746983545814230506125

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