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Properties

Label 2-39-13.6-c2-0-4
Degree $2$
Conductor $39$
Sign $-0.640 + 0.767i$
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  + (−0.816 − 3.04i)2-s + (0.866 − 1.5i)3-s + (−5.15 + 2.97i)4-s + (1.44 − 1.44i)5-s + (−5.27 − 1.41i)6-s + (−1.58 + 5.91i)7-s + (4.36 + 4.36i)8-s + (−1.5 − 2.59i)9-s + (−5.59 − 3.23i)10-s + (16.5 − 4.42i)11-s + 10.3i·12-s + (12.0 − 4.94i)13-s + 19.3·14-s + (−0.918 − 3.42i)15-s + (−2.16 + 3.75i)16-s + (−24.4 + 14.1i)17-s + ⋯
L(s)  = 1  + (−0.408 − 1.52i)2-s + (0.288 − 0.5i)3-s + (−1.28 + 0.744i)4-s + (0.289 − 0.289i)5-s + (−0.879 − 0.235i)6-s + (−0.226 + 0.844i)7-s + (0.546 + 0.546i)8-s + (−0.166 − 0.288i)9-s + (−0.559 − 0.323i)10-s + (1.50 − 0.402i)11-s + 0.859i·12-s + (0.924 − 0.380i)13-s + 1.38·14-s + (−0.0612 − 0.228i)15-s + (−0.135 + 0.234i)16-s + (−1.43 + 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ -0.640 + 0.767i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.401221 - 0.857163i\)
\(L(\frac12)\) \(\approx\) \(0.401221 - 0.857163i\)
\(L(2)\) 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 \( 1 + (-12.0 + 4.94i)T \)
good2 \( 1 + (0.816 + 3.04i)T + (-3.46 + 2i)T^{2} \)
5 \( 1 + (-1.44 + 1.44i)T - 25iT^{2} \)
7 \( 1 + (1.58 - 5.91i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-16.5 + 4.42i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (24.4 - 14.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.6 - 4.20i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (16.5 + 9.55i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.62 - 16.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (4.19 - 4.19i)T - 961iT^{2} \)
37 \( 1 + (40.5 - 10.8i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (2.32 + 8.67i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-47.3 + 27.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (10.9 + 10.9i)T + 2.20e3iT^{2} \)
53 \( 1 + 58.0T + 2.80e3T^{2} \)
59 \( 1 + (4.32 - 16.1i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (42.5 + 73.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.52 + 24.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-67.6 - 18.1i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (73.9 + 73.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 0.739T + 6.24e3T^{2} \)
83 \( 1 + (-29.6 + 29.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (102. - 27.5i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-60.9 - 16.3i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−15.56805615593219281907746695821, −13.93522212397359215027669916331, −12.84905685470703025873794808092, −11.95683116322309945405538383685, −10.88886369008489379252977106127, −9.233615554687916245643762006009, −8.685261434636637493061787180752, −6.23791029039975453731866571142, −3.57362183956764071032954922214, −1.67551174135104980136730032165, 4.26261708654315337460284569324, 6.26327211528674902318134667200, 7.24617950669371948751281824785, 8.861471773932817403790417644166, 9.723214511951638311461118942601, 11.40034834297343016450655418527, 13.75564102611775922121845351503, 14.20165080707734702612855782544, 15.55214034593229068143939822690, 16.28030321183000778028073430762

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