↑

Properties

Label 2-37-37.11-c7-0-18
Degree $2$
Conductor $37$
Sign $-0.519 + 0.854i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (18.4 − 10.6i)2-s + (13.2 − 22.9i)3-s + (164. − 284. i)4-s + (−99.3 − 57.3i)5-s − 565. i·6-s + (−72.8 + 126. i)7-s − 4.27e3i·8-s + (742. + 1.28e3i)9-s − 2.44e3·10-s − 2.49e3·11-s + (−4.34e3 − 7.52e3i)12-s + (5.11e3 + 2.95e3i)13-s + 3.11e3i·14-s + (−2.62e3 + 1.51e3i)15-s + (−2.46e4 − 4.26e4i)16-s + (2.10e4 − 1.21e4i)17-s + ⋯
L(s)  = 1  + (1.63 − 0.943i)2-s + (0.283 − 0.490i)3-s + (1.28 − 2.21i)4-s + (−0.355 − 0.205i)5-s − 1.06i·6-s + (−0.0803 + 0.139i)7-s − 2.95i·8-s + (0.339 + 0.588i)9-s − 0.774·10-s − 0.564·11-s + (−0.725 − 1.25i)12-s + (0.645 + 0.372i)13-s + 0.303i·14-s + (−0.201 + 0.116i)15-s + (−1.50 − 2.60i)16-s + (1.04 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.519 + 0.854i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.21043 - 3.92984i\)
\(L(\frac12)\) \(\approx\) \(2.21043 - 3.92984i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.92e5 - 9.59e4i)T \)
good2 \( 1 + (-18.4 + 10.6i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (-13.2 + 22.9i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (99.3 + 57.3i)T + (3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (72.8 - 126. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 2.49e3T + 1.94e7T^{2} \)
13 \( 1 + (-5.11e3 - 2.95e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-2.10e4 + 1.21e4i)T + (2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (3.33e3 + 1.92e3i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 1.04e5iT - 3.40e9T^{2} \)
29 \( 1 + 1.15e5iT - 1.72e10T^{2} \)
31 \( 1 - 3.41e3iT - 2.75e10T^{2} \)
41 \( 1 + (2.79e5 - 4.83e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 - 1.90e5iT - 2.71e11T^{2} \)
47 \( 1 - 2.22e5T + 5.06e11T^{2} \)
53 \( 1 + (-1.79e5 - 3.11e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (1.11e6 - 6.43e5i)T + (1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.95e6 + 1.12e6i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (6.72e5 - 1.16e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-1.95e6 + 3.38e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + 6.22e6T + 1.10e13T^{2} \)
79 \( 1 + (6.58e6 + 3.80e6i)T + (9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.41e6 - 2.44e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (5.76e6 - 3.32e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.30e7iT - 8.07e13T^{2} \)
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

−13.88044488722891367033027364902, −13.34082483572301886471729794563, −12.21273112634880511407838718571, −11.27368613511716483858738903283, −9.915091292453101712545603230562, −7.63799505837254944164895815479, −5.88045252264561195264710776944, −4.48462290408514285052564104589, −2.94454065707588122366176536077, −1.43252881767250378485459887325, 3.17693695253802938713382169637, 4.19175770983405132874977865391, 5.72492038174857488517169958930, 7.07663791377129799947918171758, 8.394303006879214554248702246650, 10.58290723679657410960471616584, 12.18545441402948970556930288073, 13.04517255031099372229776894789, 14.36415717287905043438015444871, 15.10418414043169499048438776969

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