Properties

Label 2-39-39.8-c3-0-7
Degree $2$
Conductor $39$
Sign $0.774 + 0.632i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.980 + 0.980i)2-s + (−2.50 − 4.55i)3-s + 6.07i·4-s + (12.5 − 12.5i)5-s + (6.91 + 2.01i)6-s + (21.4 − 21.4i)7-s + (−13.8 − 13.8i)8-s + (−14.4 + 22.7i)9-s + 24.6i·10-s + (−8.34 − 8.34i)11-s + (27.6 − 15.2i)12-s + (29.3 + 36.5i)13-s + 42.0i·14-s + (−88.5 − 25.7i)15-s − 21.5·16-s − 39.5·17-s + ⋯
L(s)  = 1  + (−0.346 + 0.346i)2-s + (−0.481 − 0.876i)3-s + 0.759i·4-s + (1.12 − 1.12i)5-s + (0.470 + 0.136i)6-s + (1.15 − 1.15i)7-s + (−0.609 − 0.609i)8-s + (−0.536 + 0.843i)9-s + 0.778i·10-s + (−0.228 − 0.228i)11-s + (0.665 − 0.365i)12-s + (0.626 + 0.779i)13-s + 0.803i·14-s + (−1.52 − 0.443i)15-s − 0.337·16-s − 0.564·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.774 + 0.632i$
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: \((2,\ 39,\ (\ :3/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06721 - 0.380652i\)
\(L(\frac12)\) \(\approx\) \(1.06721 - 0.380652i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.50 + 4.55i)T \)
13 \( 1 + (-29.3 - 36.5i)T \)
good2 \( 1 + (0.980 - 0.980i)T - 8iT^{2} \)
5 \( 1 + (-12.5 + 12.5i)T - 125iT^{2} \)
7 \( 1 + (-21.4 + 21.4i)T - 343iT^{2} \)
11 \( 1 + (8.34 + 8.34i)T + 1.33e3iT^{2} \)
17 \( 1 + 39.5T + 4.91e3T^{2} \)
19 \( 1 + (-9.15 - 9.15i)T + 6.85e3iT^{2} \)
23 \( 1 - 2.83T + 1.21e4T^{2} \)
29 \( 1 - 175. iT - 2.43e4T^{2} \)
31 \( 1 + (-95.0 - 95.0i)T + 2.97e4iT^{2} \)
37 \( 1 + (92.4 - 92.4i)T - 5.06e4iT^{2} \)
41 \( 1 + (187. - 187. i)T - 6.89e4iT^{2} \)
43 \( 1 - 52.5iT - 7.95e4T^{2} \)
47 \( 1 + (-194. - 194. i)T + 1.03e5iT^{2} \)
53 \( 1 - 473. iT - 1.48e5T^{2} \)
59 \( 1 + (395. + 395. i)T + 2.05e5iT^{2} \)
61 \( 1 - 104.T + 2.26e5T^{2} \)
67 \( 1 + (473. + 473. i)T + 3.00e5iT^{2} \)
71 \( 1 + (-313. + 313. i)T - 3.57e5iT^{2} \)
73 \( 1 + (84.4 - 84.4i)T - 3.89e5iT^{2} \)
79 \( 1 - 651.T + 4.93e5T^{2} \)
83 \( 1 + (-311. + 311. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-617. - 617. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-177. - 177. i)T + 9.12e5iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.28396234796829554129349705954, −13.90755343812215791113775385777, −13.37031771004768142804168511434, −12.18444491796858229199039168026, −10.85590063424979654054779615070, −8.933279956525260948287052005652, −7.921205125717401224790959955282, −6.55196552068897469647795789145, −4.80554747817209095987491870038, −1.39047177373774161398779838132, 2.34530753030933993145249478826, 5.29060572416295388057898225600, 6.12762973850829900301599388036, 8.776029667726074986356060271481, 9.983252736889779787760939838487, 10.79437970390487924129905009108, 11.66623383861589615926103617819, 13.86289246531885024452656654277, 15.01013805550058568555472918866, 15.38357452920869141671485898965

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