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Properties

Label 2-39-13.9-c1-0-1
Degree $2$
Conductor $39$
Sign $0.859 - 0.511i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (−1 − 1.73i)7-s − 3·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (1 − 1.73i)11-s + 12-s + (−3.5 − 0.866i)13-s + 1.99·14-s + (−0.5 + 0.866i)15-s + (0.500 − 0.866i)16-s + (3.5 + 6.06i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s − 0.447·5-s + (0.204 + 0.353i)6-s + (−0.377 − 0.654i)7-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.301 − 0.522i)11-s + 0.288·12-s + (−0.970 − 0.240i)13-s + 0.534·14-s + (−0.129 + 0.223i)15-s + (0.125 − 0.216i)16-s + (0.848 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658958 + 0.181103i\)
\(L(\frac12)\) \(\approx\) \(0.658958 + 0.181103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (3.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.67964485929451701471305897548, −15.31436868569418472237139205121, −14.25352282411028397136124411118, −12.73289175444341950296811916307, −11.86794614489910264294294396928, −10.08439462454078301203589831316, −8.363264865171006164651980684627, −7.54477102164470802446718447887, −6.24892583756641342435496526807, −3.47426894868114625087598578954, 2.86393153909535063400258284725, 5.20432204657102172725166254858, 7.22047791579711791825933304934, 9.254398779241532270121720047097, 9.791768698484293397611172234114, 11.42867064720421448825118723410, 12.14357937143744666454386491571, 14.02499101548283092459780281457, 15.27243809376413943797361286045, 15.84408544761433403349579987483

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