Properties

Label 2-65e2-1.1-c1-0-130
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  + 0.330·2-s + 2.69·3-s − 1.89·4-s + 0.890·6-s + 3.35·7-s − 1.28·8-s + 4.24·9-s + 3.24·11-s − 5.08·12-s + 1.10·14-s + 3.35·16-s + 1.94·17-s + 1.40·18-s + 1.24·19-s + 9.02·21-s + 1.07·22-s − 2.69·23-s − 3.46·24-s + 3.35·27-s − 6.33·28-s + 3·29-s − 3.78·31-s + 3.68·32-s + 8.73·33-s + 0.644·34-s − 8.02·36-s + 1.94·37-s + ⋯
L(s)  = 1  + 0.233·2-s + 1.55·3-s − 0.945·4-s + 0.363·6-s + 1.26·7-s − 0.455·8-s + 1.41·9-s + 0.978·11-s − 1.46·12-s + 0.296·14-s + 0.838·16-s + 0.472·17-s + 0.331·18-s + 0.285·19-s + 1.96·21-s + 0.228·22-s − 0.561·23-s − 0.707·24-s + 0.645·27-s − 1.19·28-s + 0.557·29-s − 0.679·31-s + 0.651·32-s + 1.52·33-s + 0.110·34-s − 1.33·36-s + 0.320·37-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

\(L(1)\) \(\approx\) \(3.940440551\)
\(L(\frac12)\) \(\approx\) \(3.940440551\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.330T + 2T^{2} \)
3 \( 1 - 2.69T + 3T^{2} \)
7 \( 1 - 3.35T + 7T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 - 1.24T + 19T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 3.78T + 31T^{2} \)
37 \( 1 - 1.94T + 37T^{2} \)
41 \( 1 + 2.78T + 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 - 2.53T + 59T^{2} \)
61 \( 1 + 7.49T + 61T^{2} \)
67 \( 1 + 4.01T + 67T^{2} \)
71 \( 1 + 5.24T + 71T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 8.61T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 5.26T + 97T^{2} \)
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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

−8.536506158843238741325359074501, −7.83815309282665750545807798012, −7.33140181362587970981948356199, −6.11354052631179696054845395312, −5.20930912779724181544462007930, −4.39035109831768457302444823897, −3.86600871743232241385164932024, −3.09159398251797629553193248040, −2.00823243963278197745698612367, −1.12408158942621849740862583341, 1.12408158942621849740862583341, 2.00823243963278197745698612367, 3.09159398251797629553193248040, 3.86600871743232241385164932024, 4.39035109831768457302444823897, 5.20930912779724181544462007930, 6.11354052631179696054845395312, 7.33140181362587970981948356199, 7.83815309282665750545807798012, 8.536506158843238741325359074501

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