↑

Properties

Label 2-37-37.10-c7-0-14
Degree $2$
Conductor $37$
Sign $0.132 + 0.991i$
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  + (−3.42 + 5.92i)2-s + (28.0 + 48.6i)3-s + (40.5 + 70.2i)4-s + (−242. − 419. i)5-s − 384.·6-s + (−642. − 1.11e3i)7-s − 1.43e3·8-s + (−485. + 840. i)9-s + 3.31e3·10-s + 763.·11-s + (−2.28e3 + 3.94e3i)12-s + (−1.83e3 − 3.17e3i)13-s + 8.79e3·14-s + (1.36e4 − 2.35e4i)15-s + (−297. + 514. i)16-s + (−1.40e4 + 2.43e4i)17-s + ⋯
L(s)  = 1  + (−0.302 + 0.523i)2-s + (0.600 + 1.04i)3-s + (0.317 + 0.549i)4-s + (−0.866 − 1.50i)5-s − 0.726·6-s + (−0.708 − 1.22i)7-s − 0.988·8-s + (−0.221 + 0.384i)9-s + 1.04·10-s + 0.173·11-s + (−0.380 + 0.659i)12-s + (−0.231 − 0.401i)13-s + 0.856·14-s + (1.04 − 1.80i)15-s + (−0.0181 + 0.0314i)16-s + (−0.694 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.434884 - 0.380797i\)
\(L(\frac12)\) \(\approx\) \(0.434884 - 0.380797i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.69e5 - 1.50e5i)T \)
good2 \( 1 + (3.42 - 5.92i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (-28.0 - 48.6i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (242. + 419. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (642. + 1.11e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 763.T + 1.94e7T^{2} \)
13 \( 1 + (1.83e3 + 3.17e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (1.40e4 - 2.43e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.51e4 + 4.35e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 1.55e4T + 3.40e9T^{2} \)
29 \( 1 - 1.09e5T + 1.72e10T^{2} \)
31 \( 1 + 1.20e5T + 2.75e10T^{2} \)
41 \( 1 + (2.41e5 + 4.19e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 5.71e5T + 2.71e11T^{2} \)
47 \( 1 - 1.19e6T + 5.06e11T^{2} \)
53 \( 1 + (-2.14e5 + 3.71e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (7.47e5 - 1.29e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (8.10e5 + 1.40e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.25e6 + 2.17e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-1.38e5 - 2.40e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 1.34e6T + 1.10e13T^{2} \)
79 \( 1 + (2.16e6 + 3.74e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (4.22e6 - 7.32e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (9.40e4 - 1.62e5i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 9.35e6T + 8.07e13T^{2} \)
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

−15.28372133623457955247080072223, −13.28074494268776811904008598262, −12.33528851105910828526550916146, −10.63802421585961319363204159324, −9.080193263582723350654128614931, −8.405241887489551791729713474719, −6.95269326703638084446325509146, −4.48573905334293580733984452215, −3.55980986522302958426022209747, −0.24535815909674039593762667825, 2.13387308785665353008231412773, 3.02521613145136345375467894165, 6.29678630644038000773340213083, 7.18069809047230107363935641141, 8.790007752486581395300667241360, 10.28248526390029299764178742078, 11.58935011555778608535825066587, 12.37864219522937808183719207685, 14.11603455569669292892369835330, 14.97805677328908376765365403462

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