↑

Properties

Label 2-13-13.4-c3-0-1
Degree $2$
Conductor $13$
Sign $0.252 + 0.967i$
Analytic cond. $0.767024$
Root an. cond. $0.875799$
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 − 1.73i)2-s + (3.5 − 6.06i)3-s + (2 + 3.46i)4-s + 13.8i·5-s + (−21 + 12.1i)6-s + (19.5 − 11.2i)7-s + 13.8i·8-s + (−11 − 19.0i)9-s + (23.9 − 41.5i)10-s + (−19.5 − 11.2i)11-s + 28.0·12-s + (−13 + 45.0i)13-s − 78·14-s + (84 + 48.4i)15-s + (39.9 − 69.2i)16-s + (−13.5 − 23.3i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.673 − 1.16i)3-s + (0.250 + 0.433i)4-s + 1.23i·5-s + (−1.42 + 0.824i)6-s + (1.05 − 0.607i)7-s + 0.612i·8-s + (−0.407 − 0.705i)9-s + (0.758 − 1.31i)10-s + (−0.534 − 0.308i)11-s + 0.673·12-s + (−0.277 + 0.960i)13-s − 1.48·14-s + (1.44 + 0.834i)15-s + (0.624 − 1.08i)16-s + (−0.192 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(0.767024\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.566359 - 0.437474i\)
\(L(\frac12)\) \(\approx\) \(0.566359 - 0.437474i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (13 - 45.0i)T \)
good2 \( 1 + (3 + 1.73i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 13.8iT - 125T^{2} \)
7 \( 1 + (-19.5 + 11.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (19.5 + 11.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (76.5 - 44.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (28.5 - 49.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-34.5 + 59.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 72.7iT - 2.97e4T^{2} \)
37 \( 1 + (34.5 + 19.9i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (340.5 + 196. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-42.5 - 73.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 342. iT - 1.03e5T^{2} \)
53 \( 1 - 426T + 1.48e5T^{2} \)
59 \( 1 + (16.5 - 9.52i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-8.5 - 14.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-142.5 - 82.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-505.5 + 291. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.00e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 + 426. iT - 5.71e5T^{2} \)
89 \( 1 + (-265.5 - 153. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.06e3 + 617. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−18.85927168084700490843340412488, −18.44175080340776231630660574013, −17.26214822321643194606012520434, −14.59910339934377349528497014381, −13.78044923184104415161506893934, −11.57781508819803023179857827817, −10.39776468923365132375400991003, −8.360288387817286688492718603371, −7.18864249379947825917987191008, −2.11014574367122488708249141919, 4.80198786544075143735648168099, 8.231085512020065232240113341638, 8.868693986454882017884086778155, 10.30179298841582890889348698943, 12.71056452343602839582459843870, 14.96920099474815101760947911563, 15.75822802309119482862262646720, 16.99354970851497277558510659429, 18.07827698741142481588090997022, 19.88636005144530613870746315734

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