↑

Properties

Label 2-39-13.4-c3-0-3
Degree $2$
Conductor $39$
Sign $0.413 - 0.910i$
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  + (4.36 + 2.52i)2-s + (−1.5 + 2.59i)3-s + (8.72 + 15.1i)4-s − 20.1i·5-s + (−13.1 + 7.56i)6-s + (−13.3 + 7.71i)7-s + 47.7i·8-s + (−4.5 − 7.79i)9-s + (50.7 − 87.9i)10-s + (23.3 + 13.4i)11-s − 52.3·12-s + (−3.96 − 46.7i)13-s − 77.8·14-s + (52.2 + 30.1i)15-s + (−50.5 + 87.5i)16-s + (11.6 + 20.1i)17-s + ⋯
L(s)  = 1  + (1.54 + 0.891i)2-s + (−0.288 + 0.499i)3-s + (1.09 + 1.88i)4-s − 1.79i·5-s + (−0.891 + 0.514i)6-s + (−0.721 + 0.416i)7-s + 2.10i·8-s + (−0.166 − 0.288i)9-s + (1.60 − 2.77i)10-s + (0.639 + 0.369i)11-s − 1.25·12-s + (−0.0845 − 0.996i)13-s − 1.48·14-s + (0.899 + 0.519i)15-s + (−0.789 + 1.36i)16-s + (0.165 + 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.413 - 0.910i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.00732 + 1.29319i\)
\(L(\frac12)\) \(\approx\) \(2.00732 + 1.29319i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.96 + 46.7i)T \)
good2 \( 1 + (-4.36 - 2.52i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 20.1iT - 125T^{2} \)
7 \( 1 + (13.3 - 7.71i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-23.3 - 13.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-11.6 - 20.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.0 - 22.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (71.0 - 122. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (1.14 - 1.98i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 37.7iT - 2.97e4T^{2} \)
37 \( 1 + (-271. - 156. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-5.08 - 2.93i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (180. + 312. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 209. iT - 1.03e5T^{2} \)
53 \( 1 - 276.T + 1.48e5T^{2} \)
59 \( 1 + (-470. + 271. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (102. + 178. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (426. + 246. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-716. + 413. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 66.1iT - 3.89e5T^{2} \)
79 \( 1 - 317.T + 4.93e5T^{2} \)
83 \( 1 - 141. iT - 5.71e5T^{2} \)
89 \( 1 + (-555. - 320. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (965. - 557. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−15.81460450956783761617889357571, −15.03167051457534373631140113294, −13.45510914175025512135770635156, −12.61387347115097425196707142210, −11.93972869815489836688473260411, −9.546132345129588413935584009692, −8.070835238840868105464823483322, −6.08254506186956173159111194341, −5.13359945187867324147800391272, −3.87908371896794475503951386736, 2.52517723789855038838594333263, 3.92728289114181318022981366204, 6.22692635749115362803275122280, 6.86026043971086118409834433336, 10.07724789007631260842628684216, 11.09515614652828434341156376399, 11.88450504717024711540831562530, 13.25809200816564607948788017369, 14.18822128832894418498280125440, 14.79986980376336269334352413802

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