Properties

Label 2-39-13.12-c3-0-0
Degree $2$
Conductor $39$
Sign $-0.927 + 0.373i$
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  + 5.21i·2-s − 3·3-s − 19.2·4-s − 5.83i·5-s − 15.6i·6-s + 31.3i·7-s − 58.6i·8-s + 9·9-s + 30.4·10-s + 16.2i·11-s + 57.7·12-s + (−43.4 + 17.5i)13-s − 163.·14-s + 17.5i·15-s + 152.·16-s + 54·17-s + ⋯
L(s)  = 1  + 1.84i·2-s − 0.577·3-s − 2.40·4-s − 0.522i·5-s − 1.06i·6-s + 1.69i·7-s − 2.59i·8-s + 0.333·9-s + 0.963·10-s + 0.446i·11-s + 1.38·12-s + (−0.927 + 0.373i)13-s − 3.11·14-s + 0.301i·15-s + 2.37·16-s + 0.770·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.927 + 0.373i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ -0.927 + 0.373i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.150293 - 0.775121i\)
\(L(\frac12)\) \(\approx\) \(0.150293 - 0.775121i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 + (43.4 - 17.5i)T \)
good2 \( 1 - 5.21iT - 8T^{2} \)
5 \( 1 + 5.83iT - 125T^{2} \)
7 \( 1 - 31.3iT - 343T^{2} \)
11 \( 1 - 16.2iT - 1.33e3T^{2} \)
17 \( 1 - 54T + 4.91e3T^{2} \)
19 \( 1 - 66.3iT - 6.85e3T^{2} \)
23 \( 1 - 182.T + 1.21e4T^{2} \)
29 \( 1 + 164.T + 2.43e4T^{2} \)
31 \( 1 + 58.9iT - 2.97e4T^{2} \)
37 \( 1 - 110. iT - 5.06e4T^{2} \)
41 \( 1 - 55.0iT - 6.89e4T^{2} \)
43 \( 1 - 113.T + 7.95e4T^{2} \)
47 \( 1 + 514. iT - 1.03e5T^{2} \)
53 \( 1 - 242.T + 1.48e5T^{2} \)
59 \( 1 - 265. iT - 2.05e5T^{2} \)
61 \( 1 + 468.T + 2.26e5T^{2} \)
67 \( 1 - 852. iT - 3.00e5T^{2} \)
71 \( 1 - 165. iT - 3.57e5T^{2} \)
73 \( 1 + 315. iT - 3.89e5T^{2} \)
79 \( 1 - 479.T + 4.93e5T^{2} \)
83 \( 1 - 574. iT - 5.71e5T^{2} \)
89 \( 1 + 66.7iT - 7.04e5T^{2} \)
97 \( 1 + 1.43e3iT - 9.12e5T^{2} \)
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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

−16.49823495160036583839356295788, −15.21837334083607554071081209374, −14.72601711292589396071019036931, −12.95640407556248556421816509067, −12.05709749225251416359284179325, −9.594210753095342392406161335151, −8.642078835470311126680523677009, −7.19687916901521482137035701600, −5.73783415054090777247628409908, −4.94541335610298128272522422069, 0.76420387168859795498402828227, 3.25494107712853535816675485353, 4.79973349865088929418985526904, 7.32923975659664106240264411862, 9.482908196067092387043079038770, 10.70373797128864100642872704438, 11.01174079882754920464175818625, 12.51716603042738158992335649175, 13.46135490029264151640893569077, 14.55763312990860560818706587702

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