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Properties

Label 2-39-13.11-c2-0-1
Degree $2$
Conductor $39$
Sign $0.851 - 0.523i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.247 + 0.924i)2-s + (−0.866 − 1.5i)3-s + (2.67 + 1.54i)4-s + (5.21 + 5.21i)5-s + (1.60 − 0.428i)6-s + (−2.33 − 8.72i)7-s + (−4.79 + 4.79i)8-s + (−1.5 + 2.59i)9-s + (−6.10 + 3.52i)10-s + (−13.9 − 3.74i)11-s − 5.34i·12-s + (6.62 − 11.1i)13-s + 8.63·14-s + (3.30 − 12.3i)15-s + (2.92 + 5.07i)16-s + (−1.28 − 0.742i)17-s + ⋯
L(s)  = 1  + (−0.123 + 0.462i)2-s + (−0.288 − 0.5i)3-s + (0.667 + 0.385i)4-s + (1.04 + 1.04i)5-s + (0.266 − 0.0714i)6-s + (−0.333 − 1.24i)7-s + (−0.599 + 0.599i)8-s + (−0.166 + 0.288i)9-s + (−0.610 + 0.352i)10-s + (−1.27 − 0.340i)11-s − 0.445i·12-s + (0.509 − 0.860i)13-s + 0.617·14-s + (0.220 − 0.822i)15-s + (0.183 + 0.316i)16-s + (−0.0756 − 0.0436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.851 - 0.523i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.851 - 0.523i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05914 + 0.299450i\)
\(L(\frac12)\) \(\approx\) \(1.05914 + 0.299450i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 + (-6.62 + 11.1i)T \)
good2 \( 1 + (0.247 - 0.924i)T + (-3.46 - 2i)T^{2} \)
5 \( 1 + (-5.21 - 5.21i)T + 25iT^{2} \)
7 \( 1 + (2.33 + 8.72i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (13.9 + 3.74i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (1.28 + 0.742i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (23.0 - 6.16i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-16.8 + 9.73i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (10.4 + 18.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-27.4 - 27.4i)T + 961iT^{2} \)
37 \( 1 + (-4.23 - 1.13i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-0.229 + 0.855i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-32.9 - 19.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (28.9 - 28.9i)T - 2.20e3iT^{2} \)
53 \( 1 - 14.1T + 2.80e3T^{2} \)
59 \( 1 + (-6.49 - 24.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (20.9 - 36.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.91 - 14.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (4.50 - 1.20i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (7.82 - 7.82i)T - 5.32e3iT^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 + (50.7 + 50.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-82.8 - 22.2i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (71.3 - 19.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−16.29537968871689192098559822048, −15.00339342750288080075922775455, −13.68500509929037164582222579137, −12.85004819674893356514689634325, −10.86247336079750060426986436291, −10.43017627696123386141052056740, −8.040767165938691909339371245142, −6.88425018168525994849957676679, −5.95982108526250435752845165636, −2.79778612842431276013033908534, 2.24419999336455408904465669134, 5.18822426000018497585025072046, 6.24408346412871547384512598894, 8.867435177436740218312630319026, 9.721484411415768355576955487110, 10.96626844496367706598382826553, 12.29814567036294520522489200453, 13.21467483014268430615184454362, 15.10071717836549372095301858352, 15.88251262330052878829470004457

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