↑

Properties

Label 2-37-37.36-c9-0-22
Degree $2$
Conductor $37$
Sign $0.874 + 0.485i$
Analytic cond. $19.0563$
Root an. cond. $4.36535$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 39.1i·2-s + 143.·3-s − 1.02e3·4-s − 1.26e3i·5-s + 5.63e3i·6-s − 8.20e3·7-s − 2.00e4i·8-s + 991.·9-s + 4.96e4·10-s + 2.99e4·11-s − 1.47e5·12-s − 1.18e5i·13-s − 3.21e5i·14-s − 1.82e5i·15-s + 2.60e5·16-s − 5.28e5i·17-s + ⋯
L(s)  = 1  + 1.73i·2-s + 1.02·3-s − 1.99·4-s − 0.906i·5-s + 1.77i·6-s − 1.29·7-s − 1.72i·8-s + 0.0503·9-s + 1.56·10-s + 0.617·11-s − 2.04·12-s − 1.15i·13-s − 2.23i·14-s − 0.929i·15-s + 0.992·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(19.0563\)
Root analytic conductor: \(4.36535\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :9/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.961348 - 0.248807i\)
\(L(\frac12)\) \(\approx\) \(0.961348 - 0.248807i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (9.96e6 + 5.53e6i)T \)
good2 \( 1 - 39.1iT - 512T^{2} \)
3 \( 1 - 143.T + 1.96e4T^{2} \)
5 \( 1 + 1.26e3iT - 1.95e6T^{2} \)
7 \( 1 + 8.20e3T + 4.03e7T^{2} \)
11 \( 1 - 2.99e4T + 2.35e9T^{2} \)
13 \( 1 + 1.18e5iT - 1.06e10T^{2} \)
17 \( 1 + 5.28e5iT - 1.18e11T^{2} \)
19 \( 1 - 4.09e5iT - 3.22e11T^{2} \)
23 \( 1 + 7.11e5iT - 1.80e12T^{2} \)
29 \( 1 + 1.52e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.81e6iT - 2.64e13T^{2} \)
41 \( 1 + 2.41e7T + 3.27e14T^{2} \)
43 \( 1 - 2.13e7iT - 5.02e14T^{2} \)
47 \( 1 - 4.13e6T + 1.11e15T^{2} \)
53 \( 1 + 8.09e7T + 3.29e15T^{2} \)
59 \( 1 + 1.97e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.53e8iT - 1.16e16T^{2} \)
67 \( 1 + 2.74e8T + 2.72e16T^{2} \)
71 \( 1 - 2.56e8T + 4.58e16T^{2} \)
73 \( 1 - 3.67e8T + 5.88e16T^{2} \)
79 \( 1 - 4.59e7iT - 1.19e17T^{2} \)
83 \( 1 - 2.92e8T + 1.86e17T^{2} \)
89 \( 1 + 7.34e8iT - 3.50e17T^{2} \)
97 \( 1 - 5.64e8iT - 7.60e17T^{2} \)
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

−14.41404692028186700654883004722, −13.54790131737733369300881595841, −12.52811709363080865572695379091, −9.643387015571668490821089359318, −8.866952004886413483431993083603, −7.86198538645453358546657834815, −6.49124565816651396892820608311, −5.09879076842906564039021692787, −3.31257259019356377979940331755, −0.31227353432625000864390171434, 1.87207699285481007040050445163, 3.09165098203349571464788411355, 3.83087597130145370352550261433, 6.65659277633236756606296988258, 8.794166657897084816702726050695, 9.621143385776986493182554202664, 10.74197695114496667608483016119, 11.95632230079146933777277156243, 13.21856256721216946838277403234, 14.02346000569955336671235413437

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