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Properties

Label 2-39-39.35-c4-0-16
Degree $2$
Conductor $39$
Sign $-0.674 - 0.737i$
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  + (−4.18 − 2.41i)2-s + (−2.70 − 8.58i)3-s + (3.66 + 6.34i)4-s − 24.8i·5-s + (−9.40 + 42.4i)6-s + (−4.36 − 7.55i)7-s + 41.8i·8-s + (−66.3 + 46.4i)9-s + (−59.9 + 103. i)10-s + (44.8 + 25.8i)11-s + (44.5 − 48.6i)12-s + (−128. + 109. i)13-s + 42.1i·14-s + (−212. + 67.1i)15-s + (159. − 276. i)16-s + (135. − 78.3i)17-s + ⋯
L(s)  = 1  + (−1.04 − 0.603i)2-s + (−0.300 − 0.953i)3-s + (0.228 + 0.396i)4-s − 0.992i·5-s + (−0.261 + 1.17i)6-s + (−0.0890 − 0.154i)7-s + 0.654i·8-s + (−0.818 + 0.573i)9-s + (−0.599 + 1.03i)10-s + (0.370 + 0.213i)11-s + (0.309 − 0.337i)12-s + (−0.762 + 0.646i)13-s + 0.215i·14-s + (−0.946 + 0.298i)15-s + (0.624 − 1.08i)16-s + (0.469 − 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.737i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.674 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.674 - 0.737i$
Analytic conductor: \(4.03142\)
Root analytic conductor: \(2.00784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :2),\ -0.674 - 0.737i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.152103 + 0.345247i\)
\(L(\frac12)\) \(\approx\) \(0.152103 + 0.345247i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.70 + 8.58i)T \)
13 \( 1 + (128. - 109. i)T \)
good2 \( 1 + (4.18 + 2.41i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + 24.8iT - 625T^{2} \)
7 \( 1 + (4.36 + 7.55i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-44.8 - 25.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
17 \( 1 + (-135. + 78.3i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (237. + 411. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-36.3 - 20.9i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (846. + 488. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.48e3T + 9.23e5T^{2} \)
37 \( 1 + (179. - 310. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (2.42e3 + 1.40e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.21e3 - 2.10e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + 172. iT - 4.87e6T^{2} \)
53 \( 1 + 3.79e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.36e3 + 1.36e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-140. - 242. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.73e3 + 2.99e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-185. + 106. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + 7.88e3T + 2.83e7T^{2} \)
79 \( 1 - 1.05e4T + 3.89e7T^{2} \)
83 \( 1 - 9.49e3iT - 4.74e7T^{2} \)
89 \( 1 + (9.34e3 + 5.39e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (4.27e3 + 7.40e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−14.57364499518708375225720704547, −13.19741752176308722962507571468, −12.11282514250396384689114437915, −11.14009561304777524024168141255, −9.530437640301700513326758666313, −8.560228518997041230154567761653, −7.15211006360020739377705631756, −5.16153435061259792023411617692, −1.91727212063535460937212860522, −0.36365959075158134046458836578, 3.56750121295011115551553682767, 5.87913566609701120008237458037, 7.33104836875850195604872863283, 8.806994069452134944031447684755, 10.01016713400296996944172552957, 10.79138610745561959215689752247, 12.43792806855238342251524598993, 14.55669303120724456234396278532, 15.17960590314150230886468655475, 16.48455038768196230343353038633

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