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Properties

Label 2-39-13.10-c3-0-2
Degree $2$
Conductor $39$
Sign $0.252 - 0.967i$
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  + (1.5 + 2.59i)3-s + (−4 + 6.92i)4-s + 5.19i·5-s + (9 + 5.19i)7-s + (−4.5 + 7.79i)9-s + (45 − 25.9i)11-s − 24·12-s + (−32.5 + 33.7i)13-s + (−13.5 + 7.79i)15-s + (−31.9 − 55.4i)16-s + (58.5 − 101. i)17-s + (−21 − 12.1i)19-s + (−36 − 20.7i)20-s + 31.1i·21-s + (−9 − 15.5i)23-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.5 + 0.866i)4-s + 0.464i·5-s + (0.485 + 0.280i)7-s + (−0.166 + 0.288i)9-s + (1.23 − 0.712i)11-s − 0.577·12-s + (−0.693 + 0.720i)13-s + (−0.232 + 0.134i)15-s + (−0.499 − 0.866i)16-s + (0.834 − 1.44i)17-s + (−0.253 − 0.146i)19-s + (−0.402 − 0.232i)20-s + 0.323i·21-s + (−0.0815 − 0.141i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03450 + 0.799085i\)
\(L(\frac12)\) \(\approx\) \(1.03450 + 0.799085i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 2.59i)T \)
13 \( 1 + (32.5 - 33.7i)T \)
good2 \( 1 + (4 - 6.92i)T^{2} \)
5 \( 1 - 5.19iT - 125T^{2} \)
7 \( 1 + (-9 - 5.19i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-45 + 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-58.5 + 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21 + 12.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (9 + 15.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-49.5 - 85.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 193. iT - 2.97e4T^{2} \)
37 \( 1 + (-97.5 + 56.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (31.5 - 18.1i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-41 + 71.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 72.7iT - 1.03e5T^{2} \)
53 \( 1 + 261T + 1.48e5T^{2} \)
59 \( 1 + (684 + 394. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-359.5 + 622. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (609 - 351. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (405 + 233. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 684. iT - 3.89e5T^{2} \)
79 \( 1 + 440T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 + (-1.31e3 + 758. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.00e3 + 578. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−16.19070696740159999958382328021, −14.46042472992647119789468202108, −14.04472311041188170855034220425, −12.25832657975772405799141516690, −11.30467924244811007145418504708, −9.525540148482312574959661327584, −8.574378121638728846891101503349, −7.04493677831332736931288409360, −4.77684931404360050921121243664, −3.17206571869895288643449316524, 1.36087313542375749807834651046, 4.42164068468694508024683721833, 6.09233237928203958508283199163, 7.86280362991122280341051141812, 9.225365385306053188226397977005, 10.40673653615288329941184404978, 12.09929814719852620392873192070, 13.17085211965414627495141601243, 14.54893601012994124085389315948, 14.95312518381551538990764003931

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