Properties

Label 2-39-1.1-c3-0-4
Degree $2$
Conductor $39$
Sign $1$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.20·2-s + 3·3-s + 9.71·4-s − 11.4·5-s + 12.6·6-s − 11.2·7-s + 7.22·8-s + 9·9-s − 48.1·10-s + 25.8·11-s + 29.1·12-s + 13·13-s − 47.3·14-s − 34.2·15-s − 47.3·16-s − 20.3·17-s + 37.8·18-s + 154.·19-s − 111.·20-s − 33.7·21-s + 108.·22-s − 180.·23-s + 21.6·24-s + 5.69·25-s + 54.7·26-s + 27·27-s − 109.·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.577·3-s + 1.21·4-s − 1.02·5-s + 0.859·6-s − 0.607·7-s + 0.319·8-s + 0.333·9-s − 1.52·10-s + 0.709·11-s + 0.701·12-s + 0.277·13-s − 0.904·14-s − 0.590·15-s − 0.739·16-s − 0.290·17-s + 0.496·18-s + 1.86·19-s − 1.24·20-s − 0.350·21-s + 1.05·22-s − 1.63·23-s + 0.184·24-s + 0.0455·25-s + 0.412·26-s + 0.192·27-s − 0.738·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $1$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.527629675\)
\(L(\frac12)\) \(\approx\) \(2.527629675\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 - 13T \)
good2 \( 1 - 4.20T + 8T^{2} \)
5 \( 1 + 11.4T + 125T^{2} \)
7 \( 1 + 11.2T + 343T^{2} \)
11 \( 1 - 25.8T + 1.33e3T^{2} \)
17 \( 1 + 20.3T + 4.91e3T^{2} \)
19 \( 1 - 154.T + 6.85e3T^{2} \)
23 \( 1 + 180.T + 1.21e4T^{2} \)
29 \( 1 + 20.4T + 2.43e4T^{2} \)
31 \( 1 - 266.T + 2.97e4T^{2} \)
37 \( 1 - 115.T + 5.06e4T^{2} \)
41 \( 1 - 391.T + 6.89e4T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 + 467.T + 1.03e5T^{2} \)
53 \( 1 - 79.9T + 1.48e5T^{2} \)
59 \( 1 + 873.T + 2.05e5T^{2} \)
61 \( 1 + 187.T + 2.26e5T^{2} \)
67 \( 1 + 609.T + 3.00e5T^{2} \)
71 \( 1 - 248.T + 3.57e5T^{2} \)
73 \( 1 - 852.T + 3.89e5T^{2} \)
79 \( 1 + 331.T + 4.93e5T^{2} \)
83 \( 1 + 435.T + 5.71e5T^{2} \)
89 \( 1 - 259.T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
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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

−15.61092336644773112190771981736, −14.35050901270024830458836863717, −13.57424298072454794785598028302, −12.28893091620450122361463384680, −11.52880811330999102287071951704, −9.502694959866380525191190996779, −7.76775780652918002097914124192, −6.23769169350368111799495865864, −4.30788066429207028148735388895, −3.22991637083297119472160624174, 3.22991637083297119472160624174, 4.30788066429207028148735388895, 6.23769169350368111799495865864, 7.76775780652918002097914124192, 9.502694959866380525191190996779, 11.52880811330999102287071951704, 12.28893091620450122361463384680, 13.57424298072454794785598028302, 14.35050901270024830458836863717, 15.61092336644773112190771981736

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