↑

Properties

Label 2-37-37.36-c5-0-6
Degree $2$
Conductor $37$
Sign $0.988 + 0.148i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.32i·2-s − 24.1·3-s + 13.3·4-s − 61.4i·5-s − 104. i·6-s + 41.8·7-s + 195. i·8-s + 341.·9-s + 265.·10-s + 646.·11-s − 322.·12-s − 842. i·13-s + 180. i·14-s + 1.48e3i·15-s − 420.·16-s − 1.32e3i·17-s + ⋯
L(s)  = 1  + 0.763i·2-s − 1.55·3-s + 0.416·4-s − 1.10i·5-s − 1.18i·6-s + 0.322·7-s + 1.08i·8-s + 1.40·9-s + 0.840·10-s + 1.61·11-s − 0.645·12-s − 1.38i·13-s + 0.246i·14-s + 1.70i·15-s − 0.410·16-s − 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.988 + 0.148i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.16386 - 0.0871938i\)
\(L(\frac12)\) \(\approx\) \(1.16386 - 0.0871938i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (8.23e3 + 1.24e3i)T \)
good2 \( 1 - 4.32iT - 32T^{2} \)
3 \( 1 + 24.1T + 243T^{2} \)
5 \( 1 + 61.4iT - 3.12e3T^{2} \)
7 \( 1 - 41.8T + 1.68e4T^{2} \)
11 \( 1 - 646.T + 1.61e5T^{2} \)
13 \( 1 + 842. iT - 3.71e5T^{2} \)
17 \( 1 + 1.32e3iT - 1.41e6T^{2} \)
19 \( 1 - 328. iT - 2.47e6T^{2} \)
23 \( 1 + 3.11e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.64e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.32e3iT - 2.86e7T^{2} \)
41 \( 1 - 7.56e3T + 1.15e8T^{2} \)
43 \( 1 + 8.90e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.00e4T + 2.29e8T^{2} \)
53 \( 1 + 1.28e3T + 4.18e8T^{2} \)
59 \( 1 - 3.48e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.44e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.23e3T + 1.35e9T^{2} \)
71 \( 1 - 5.72e4T + 1.80e9T^{2} \)
73 \( 1 + 5.22e3T + 2.07e9T^{2} \)
79 \( 1 + 5.39e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.43e4T + 3.93e9T^{2} \)
89 \( 1 + 6.49e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.14e4iT - 8.58e9T^{2} \)
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

−15.80510663945102472295309390372, −14.41936678411814708783291310553, −12.52795284632579358406334658958, −11.80612517165895341407169735472, −10.66720638527729214884497782131, −8.766094402020946980244437047577, −7.09642680604868634469673539023, −5.84183442252684146046610063642, −4.86501931537900095269544943329, −0.925585043322006805173697215167, 1.55942477274422211339483823576, 3.97082748222637845587782205274, 6.25504846235333942824979136529, 6.90684167126372906040149930416, 9.706538196779585658461950279503, 11.02621674602494928259077697329, 11.43649616264418843503208643548, 12.30404565952378413870241436477, 14.18669267910261158443913023780, 15.52020442779562100751835465235

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