↑

Properties

Label 2-39-13.10-c3-0-5
Degree $2$
Conductor $39$
Sign $0.842 + 0.538i$
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  + (3.57 − 2.06i)2-s + (1.5 + 2.59i)3-s + (4.50 − 7.79i)4-s − 3.05i·5-s + (10.7 + 6.18i)6-s + (−5.78 − 3.34i)7-s − 4.12i·8-s + (−4.5 + 7.79i)9-s + (−6.28 − 10.8i)10-s + (−27.9 + 16.1i)11-s + 27.0·12-s + (−22.1 − 41.3i)13-s − 27.5·14-s + (7.92 − 4.57i)15-s + (27.4 + 47.6i)16-s + (−14.4 + 24.9i)17-s + ⋯
L(s)  = 1  + (1.26 − 0.728i)2-s + (0.288 + 0.499i)3-s + (0.562 − 0.974i)4-s − 0.272i·5-s + (0.728 + 0.420i)6-s + (−0.312 − 0.180i)7-s − 0.182i·8-s + (−0.166 + 0.288i)9-s + (−0.198 − 0.344i)10-s + (−0.765 + 0.441i)11-s + 0.649·12-s + (−0.472 − 0.881i)13-s − 0.526·14-s + (0.136 − 0.0787i)15-s + (0.429 + 0.744i)16-s + (−0.205 + 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.842 + 0.538i$
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: \((2,\ 39,\ (\ :3/2),\ 0.842 + 0.538i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.23466 - 0.652462i\)
\(L(\frac12)\) \(\approx\) \(2.23466 - 0.652462i\)
\(L(\frac{5}{2})\) 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 \( 1 + (22.1 + 41.3i)T \)
good2 \( 1 + (-3.57 + 2.06i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 3.05iT - 125T^{2} \)
7 \( 1 + (5.78 + 3.34i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (27.9 - 16.1i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (14.4 - 24.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-87.6 - 50.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (59.4 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (80.0 + 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 38.0iT - 2.97e4T^{2} \)
37 \( 1 + (-283. + 163. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-48.5 + 28.0i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-63.9 + 110. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 517. iT - 1.03e5T^{2} \)
53 \( 1 + 695.T + 1.48e5T^{2} \)
59 \( 1 + (-568. - 328. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (350. - 607. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (49.5 - 28.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (267. + 154. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 389. iT - 3.89e5T^{2} \)
79 \( 1 - 901.T + 4.93e5T^{2} \)
83 \( 1 - 687. iT - 5.71e5T^{2} \)
89 \( 1 + (927. - 535. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.51e3 + 877. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−15.25637736073738160766511101110, −14.32680304512919641170116703348, −13.11245334490740739837074166361, −12.38309530084941604137348254434, −10.90278905675412221686278518675, −9.834682235292705628733824368901, −7.955262631434725645731715206078, −5.62503951100183094920402058110, −4.32761933821869543892968411129, −2.76113370361538354469163619614, 3.11429660786754407067721342564, 5.07111828248195752304953152354, 6.52590347559654692379567273262, 7.60187918493724442768531567253, 9.476045523362915230513697705406, 11.47574450689179592881531640645, 12.74454881657731490412515082260, 13.65502986646947209428845529494, 14.47455915233071643916205065923, 15.62526117404837519491671176704

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