Properties

Label 2-4675-1.1-c1-0-46
Degree $2$
Conductor $4675$
Sign $1$
Analytic cond. $37.3300$
Root an. cond. $6.10983$
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  + 1.78·2-s − 1.50·3-s + 1.18·4-s − 2.69·6-s − 1.32·7-s − 1.45·8-s − 0.724·9-s − 11-s − 1.78·12-s + 2.54·13-s − 2.36·14-s − 4.96·16-s + 17-s − 1.29·18-s − 5.38·19-s + 2·21-s − 1.78·22-s − 2.64·23-s + 2.19·24-s + 4.53·26-s + 5.61·27-s − 1.56·28-s + 0.584·29-s + 0.916·31-s − 5.94·32-s + 1.50·33-s + 1.78·34-s + ⋯
L(s)  = 1  + 1.26·2-s − 0.870·3-s + 0.591·4-s − 1.09·6-s − 0.501·7-s − 0.515·8-s − 0.241·9-s − 0.301·11-s − 0.514·12-s + 0.704·13-s − 0.632·14-s − 1.24·16-s + 0.242·17-s − 0.304·18-s − 1.23·19-s + 0.436·21-s − 0.380·22-s − 0.550·23-s + 0.448·24-s + 0.889·26-s + 1.08·27-s − 0.296·28-s + 0.108·29-s + 0.164·31-s − 1.05·32-s + 0.262·33-s + 0.305·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4675\)    =    \(5^{2} \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(37.3300\)
Root analytic conductor: \(6.10983\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.668907440\)
\(L(\frac12)\) \(\approx\) \(1.668907440\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - 1.78T + 2T^{2} \)
3 \( 1 + 1.50T + 3T^{2} \)
7 \( 1 + 1.32T + 7T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 - 0.584T + 29T^{2} \)
31 \( 1 - 0.916T + 31T^{2} \)
37 \( 1 - 8.04T + 37T^{2} \)
41 \( 1 - 2.54T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 - 5.95T + 53T^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 - 3.20T + 61T^{2} \)
67 \( 1 - 9.91T + 67T^{2} \)
71 \( 1 + 9.13T + 71T^{2} \)
73 \( 1 + 1.91T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 19.6T + 97T^{2} \)
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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

−8.376932118853216345711914098025, −7.26589161341369199360278824156, −6.28107975508800047382231358093, −6.12407417512236884685829026776, −5.42534021880149405088602261404, −4.60426164329440811003056724147, −3.97105051041924211335749107578, −3.10884724330932900568878098744, −2.25841631077319933590654974365, −0.59125606293068858768890167400, 0.59125606293068858768890167400, 2.25841631077319933590654974365, 3.10884724330932900568878098744, 3.97105051041924211335749107578, 4.60426164329440811003056724147, 5.42534021880149405088602261404, 6.12407417512236884685829026776, 6.28107975508800047382231358093, 7.26589161341369199360278824156, 8.376932118853216345711914098025

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