Properties

Label 2-39-1.1-c1-0-1
Degree $2$
Conductor $39$
Sign $1$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
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  + 0.414·2-s + 3-s − 1.82·4-s − 2.82·5-s + 0.414·6-s + 2.82·7-s − 1.58·8-s + 9-s − 1.17·10-s − 2·11-s − 1.82·12-s − 13-s + 1.17·14-s − 2.82·15-s + 3·16-s + 7.65·17-s + 0.414·18-s − 2.82·19-s + 5.17·20-s + 2.82·21-s − 0.828·22-s − 4·23-s − 1.58·24-s + 3.00·25-s − 0.414·26-s + 27-s − 5.17·28-s + ⋯
L(s)  = 1  + 0.292·2-s + 0.577·3-s − 0.914·4-s − 1.26·5-s + 0.169·6-s + 1.06·7-s − 0.560·8-s + 0.333·9-s − 0.370·10-s − 0.603·11-s − 0.527·12-s − 0.277·13-s + 0.313·14-s − 0.730·15-s + 0.750·16-s + 1.85·17-s + 0.0976·18-s − 0.648·19-s + 1.15·20-s + 0.617·21-s − 0.176·22-s − 0.834·23-s − 0.323·24-s + 0.600·25-s − 0.0812·26-s + 0.192·27-s − 0.977·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7964221304\)
\(L(\frac12)\) \(\approx\) \(0.7964221304\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 - 0.414T + 2T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−16.04950298307033131008500397708, −14.79138962396656964896705769639, −14.25172602711280113378671227511, −12.75072538387550213063268470792, −11.73521024606481280324460127322, −10.09067269620032594182897164525, −8.367796056521928312460730790959, −7.79055936858715051043363136838, −5.06473968936367901043856637263, −3.71045824363177121690250957726, 3.71045824363177121690250957726, 5.06473968936367901043856637263, 7.79055936858715051043363136838, 8.367796056521928312460730790959, 10.09067269620032594182897164525, 11.73521024606481280324460127322, 12.75072538387550213063268470792, 14.25172602711280113378671227511, 14.79138962396656964896705769639, 16.04950298307033131008500397708

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