Properties

Label 2-4675-1.1-c1-0-127
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  + 1.21·2-s + 2.36·3-s − 0.515·4-s + 2.87·6-s + 0.846·7-s − 3.06·8-s + 2.58·9-s − 11-s − 1.21·12-s + 0.935·13-s + 1.03·14-s − 2.70·16-s + 17-s + 3.14·18-s + 5.75·19-s + 2·21-s − 1.21·22-s − 2.54·23-s − 7.24·24-s + 1.13·26-s − 0.990·27-s − 0.436·28-s + 9.45·29-s + 4.12·31-s + 2.83·32-s − 2.36·33-s + 1.21·34-s + ⋯
L(s)  = 1  + 0.861·2-s + 1.36·3-s − 0.257·4-s + 1.17·6-s + 0.319·7-s − 1.08·8-s + 0.860·9-s − 0.301·11-s − 0.351·12-s + 0.259·13-s + 0.275·14-s − 0.675·16-s + 0.242·17-s + 0.741·18-s + 1.32·19-s + 0.436·21-s − 0.259·22-s − 0.531·23-s − 1.47·24-s + 0.223·26-s − 0.190·27-s − 0.0825·28-s + 1.75·29-s + 0.741·31-s + 0.501·32-s − 0.411·33-s + 0.208·34-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

\(L(1)\) \(\approx\) \(4.501001345\)
\(L(\frac12)\) \(\approx\) \(4.501001345\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.21T + 2T^{2} \)
3 \( 1 - 2.36T + 3T^{2} \)
7 \( 1 - 0.846T + 7T^{2} \)
13 \( 1 - 0.935T + 13T^{2} \)
19 \( 1 - 5.75T + 19T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 - 9.45T + 29T^{2} \)
31 \( 1 - 4.12T + 31T^{2} \)
37 \( 1 - 0.776T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 - 7.94T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 9.72T + 61T^{2} \)
67 \( 1 + 1.13T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 - 9.13T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 9.10T + 83T^{2} \)
89 \( 1 + 0.835T + 89T^{2} \)
97 \( 1 + 8.21T + 97T^{2} \)
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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

−8.227305018659894591812886978055, −7.84534792311009503447869099393, −6.87996653860080187659039300898, −5.93613894518723919906825109512, −5.22683762668709111034486538608, −4.41824722826831634403782856247, −3.73856900327122958065009695992, −2.97534975684993375382473718888, −2.42345514140626795396446306676, −1.00834730735439999590031028305, 1.00834730735439999590031028305, 2.42345514140626795396446306676, 2.97534975684993375382473718888, 3.73856900327122958065009695992, 4.41824722826831634403782856247, 5.22683762668709111034486538608, 5.93613894518723919906825109512, 6.87996653860080187659039300898, 7.84534792311009503447869099393, 8.227305018659894591812886978055

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