↑

Properties

Label 2-13e2-13.4-c3-0-6
Degree $2$
Conductor $169$
Sign $0.425 - 0.905i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
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  + (0.337 + 0.195i)2-s + (−1.80 + 3.12i)3-s + (−3.92 − 6.79i)4-s − 7.52i·5-s + (−1.21 + 0.704i)6-s + (−16.9 + 9.77i)7-s − 6.18i·8-s + (6.98 + 12.0i)9-s + (1.46 − 2.54i)10-s + (39.6 + 22.9i)11-s + 28.3·12-s − 7.62·14-s + (23.5 + 13.5i)15-s + (−30.1 + 52.2i)16-s + (43.2 + 74.9i)17-s + 5.45i·18-s + ⋯
L(s)  = 1  + (0.119 + 0.0689i)2-s + (−0.347 + 0.601i)3-s + (−0.490 − 0.849i)4-s − 0.672i·5-s + (−0.0829 + 0.0479i)6-s + (−0.913 + 0.527i)7-s − 0.273i·8-s + (0.258 + 0.448i)9-s + (0.0463 − 0.0803i)10-s + (1.08 + 0.628i)11-s + 0.681·12-s − 0.145·14-s + (0.404 + 0.233i)15-s + (−0.471 + 0.816i)16-s + (0.617 + 1.06i)17-s + 0.0713i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.425 - 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03250 + 0.655566i\)
\(L(\frac12)\) \(\approx\) \(1.03250 + 0.655566i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.337 - 0.195i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (1.80 - 3.12i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 7.52iT - 125T^{2} \)
7 \( 1 + (16.9 - 9.77i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-39.6 - 22.9i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-43.2 - 74.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-128. + 74.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (45.7 - 79.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (129. - 224. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 31.2iT - 2.97e4T^{2} \)
37 \( 1 + (-128. - 74.2i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (83.0 + 47.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (40.4 + 70.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 94.3iT - 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 + (-498. + 287. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-20.1 - 34.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (520. + 300. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (449. - 259. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 + 320.T + 4.93e5T^{2} \)
83 \( 1 + 32.4iT - 5.71e5T^{2} \)
89 \( 1 + (-390. - 225. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-200. + 115. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−12.60583125380083809319198947814, −11.49524429842398952160424949159, −10.21624050072589082658442632422, −9.555687818512092669573611659339, −8.853258503313090926294082542624, −7.06269484793321835595421612614, −5.73065281583456331885976795218, −4.98929772623156870658772502351, −3.76501718058066037251765335805, −1.35626336953515640914905652999, 0.66496576014329459262197935786, 3.11666668484822466404595508994, 3.96227371482619572436242875053, 5.91538181689369218219263246240, 6.95177539320177754393417459849, 7.68498342405009089147977226886, 9.221811497066639105148881943861, 9.989191301611925980383192224616, 11.64692665559258183603974127466, 11.99078707670598668059810717224

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