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Properties

Label 2-39-13.5-c4-0-2
Degree $2$
Conductor $39$
Sign $0.348 - 0.937i$
Analytic cond. $4.03142$
Root an. cond. $2.00784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0868 − 0.0868i)2-s + 5.19·3-s + 15.9i·4-s + (−14.6 + 14.6i)5-s + (0.451 − 0.451i)6-s + (21.4 + 21.4i)7-s + (2.77 + 2.77i)8-s + 27·9-s + 2.53i·10-s + (21.6 + 21.6i)11-s + 83.0i·12-s + (134. + 102. i)13-s + 3.73·14-s + (−75.9 + 75.9i)15-s − 255.·16-s − 251. i·17-s + ⋯
L(s)  = 1  + (0.0217 − 0.0217i)2-s + 0.577·3-s + 0.999i·4-s + (−0.584 + 0.584i)5-s + (0.0125 − 0.0125i)6-s + (0.438 + 0.438i)7-s + (0.0434 + 0.0434i)8-s + 0.333·9-s + 0.0253i·10-s + (0.179 + 0.179i)11-s + 0.576i·12-s + (0.795 + 0.605i)13-s + 0.0190·14-s + (−0.337 + 0.337i)15-s − 0.997·16-s − 0.871i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(4.03142\)
Root analytic conductor: \(2.00784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :2),\ 0.348 - 0.937i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.35302 + 0.940244i\)
\(L(\frac12)\) \(\approx\) \(1.35302 + 0.940244i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
13 \( 1 + (-134. - 102. i)T \)
good2 \( 1 + (-0.0868 + 0.0868i)T - 16iT^{2} \)
5 \( 1 + (14.6 - 14.6i)T - 625iT^{2} \)
7 \( 1 + (-21.4 - 21.4i)T + 2.40e3iT^{2} \)
11 \( 1 + (-21.6 - 21.6i)T + 1.46e4iT^{2} \)
17 \( 1 + 251. iT - 8.35e4T^{2} \)
19 \( 1 + (-133. + 133. i)T - 1.30e5iT^{2} \)
23 \( 1 + 1.02e3iT - 2.79e5T^{2} \)
29 \( 1 + 479.T + 7.07e5T^{2} \)
31 \( 1 + (-359. + 359. i)T - 9.23e5iT^{2} \)
37 \( 1 + (-1.22e3 - 1.22e3i)T + 1.87e6iT^{2} \)
41 \( 1 + (-381. + 381. i)T - 2.82e6iT^{2} \)
43 \( 1 - 356. iT - 3.41e6T^{2} \)
47 \( 1 + (-2.00e3 - 2.00e3i)T + 4.87e6iT^{2} \)
53 \( 1 - 1.71e3T + 7.89e6T^{2} \)
59 \( 1 + (232. + 232. i)T + 1.21e7iT^{2} \)
61 \( 1 + 1.45e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.50e3 + 4.50e3i)T - 2.01e7iT^{2} \)
71 \( 1 + (4.29e3 - 4.29e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (6.22e3 + 6.22e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 321.T + 3.89e7T^{2} \)
83 \( 1 + (8.93e3 - 8.93e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (-5.66e3 - 5.66e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (3.99e3 - 3.99e3i)T - 8.85e7iT^{2} \)
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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

−15.66115915149284555370334502657, −14.54971757133000439068174204341, −13.38000834006096860564412880471, −12.03522110395902142172404974694, −11.11600276336256012567801257159, −9.140720652017813243236583735731, −8.062036568779736537048369583420, −6.87811751329199377455761564488, −4.27155658081843551028971862461, −2.74835389256305901921531214828, 1.23955271604401816564576673670, 4.00161297352528506863309227174, 5.70620429260033443653070208152, 7.64947798495812653899538525312, 8.891895497767368562166555866924, 10.28326658885216563860789137543, 11.48541575336187640964691707583, 13.12137926336013072495884417515, 14.14481805702882359206635249090, 15.23532702175286788063731785175

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