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Properties

Label 2-39-39.8-c3-0-0
Degree $2$
Conductor $39$
Sign $-0.957 - 0.288i$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.05 + 2.05i)2-s + (0.159 − 5.19i)3-s − 0.445i·4-s + (−14.0 + 14.0i)5-s + (10.3 + 11.0i)6-s + (−5.62 + 5.62i)7-s + (−15.5 − 15.5i)8-s + (−26.9 − 1.65i)9-s − 57.7i·10-s + (25.0 + 25.0i)11-s + (−2.31 − 0.0709i)12-s + (−46.8 + 1.50i)13-s − 23.1i·14-s + (70.6 + 75.1i)15-s + 67.3·16-s + 82.2·17-s + ⋯
L(s)  = 1  + (−0.726 + 0.726i)2-s + (0.0306 − 0.999i)3-s − 0.0556i·4-s + (−1.25 + 1.25i)5-s + (0.703 + 0.748i)6-s + (−0.303 + 0.303i)7-s + (−0.686 − 0.686i)8-s + (−0.998 − 0.0613i)9-s − 1.82i·10-s + (0.687 + 0.687i)11-s + (−0.0556 − 0.00170i)12-s + (−0.999 + 0.0320i)13-s − 0.441i·14-s + (1.21 + 1.29i)15-s + 1.05·16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.957 - 0.288i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.957 - 0.288i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0558112 + 0.378680i\)
\(L(\frac12)\) \(\approx\) \(0.0558112 + 0.378680i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.159 + 5.19i)T \)
13 \( 1 + (46.8 - 1.50i)T \)
good2 \( 1 + (2.05 - 2.05i)T - 8iT^{2} \)
5 \( 1 + (14.0 - 14.0i)T - 125iT^{2} \)
7 \( 1 + (5.62 - 5.62i)T - 343iT^{2} \)
11 \( 1 + (-25.0 - 25.0i)T + 1.33e3iT^{2} \)
17 \( 1 - 82.2T + 4.91e3T^{2} \)
19 \( 1 + (-29.4 - 29.4i)T + 6.85e3iT^{2} \)
23 \( 1 + 107.T + 1.21e4T^{2} \)
29 \( 1 - 29.9iT - 2.43e4T^{2} \)
31 \( 1 + (-62.0 - 62.0i)T + 2.97e4iT^{2} \)
37 \( 1 + (132. - 132. i)T - 5.06e4iT^{2} \)
41 \( 1 + (238. - 238. i)T - 6.89e4iT^{2} \)
43 \( 1 - 140. iT - 7.95e4T^{2} \)
47 \( 1 + (172. + 172. i)T + 1.03e5iT^{2} \)
53 \( 1 - 94.0iT - 1.48e5T^{2} \)
59 \( 1 + (78.3 + 78.3i)T + 2.05e5iT^{2} \)
61 \( 1 - 152.T + 2.26e5T^{2} \)
67 \( 1 + (-287. - 287. i)T + 3.00e5iT^{2} \)
71 \( 1 + (188. - 188. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-708. + 708. i)T - 3.89e5iT^{2} \)
79 \( 1 + 603.T + 4.93e5T^{2} \)
83 \( 1 + (-437. + 437. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-586. - 586. i)T + 7.04e5iT^{2} \)
97 \( 1 + (496. + 496. i)T + 9.12e5iT^{2} \)
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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

−16.37240468388979204267640801838, −15.11391640024556778030427651264, −14.36482413062319395881502262713, −12.28237964426718519866773376451, −11.85883224969086187804238475731, −9.881094082151780007741094850388, −8.127054481627718851087887394412, −7.35547385600035285888016876629, −6.49029362412910672498743746888, −3.23963193061035266974786255684, 0.39970543721599178306687452621, 3.67639214928507753870916081557, 5.27150898288277223480033375088, 8.091160348568270662821359980661, 9.128834893804075686229051006304, 10.10958734905389730254442347263, 11.53618377536760897622885034894, 12.17107808715825785601899494082, 14.23745863430726872022057660286, 15.47659441423794702792035814833

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