↑

Properties

Label 2-37-37.36-c7-0-13
Degree $2$
Conductor $37$
Sign $0.402 + 0.915i$
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  + 0.118i·2-s − 7.03·3-s + 127.·4-s − 225. i·5-s − 0.830i·6-s + 6.20·7-s + 30.2i·8-s − 2.13e3·9-s + 26.6·10-s + 5.45e3·11-s − 900.·12-s − 9.33e3i·13-s + 0.731i·14-s + 1.58e3i·15-s + 1.63e4·16-s − 2.42e4i·17-s + ⋯
L(s)  = 1  + 0.0104i·2-s − 0.150·3-s + 0.999·4-s − 0.806i·5-s − 0.00156i·6-s + 0.00683·7-s + 0.0208i·8-s − 0.977·9-s + 0.00841·10-s + 1.23·11-s − 0.150·12-s − 1.17i·13-s + 7.12e−5i·14-s + 0.121i·15-s + 0.999·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.65151 - 1.07832i\)
\(L(\frac12)\) \(\approx\) \(1.65151 - 1.07832i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.23e5 - 2.82e5i)T \)
good2 \( 1 - 0.118iT - 128T^{2} \)
3 \( 1 + 7.03T + 2.18e3T^{2} \)
5 \( 1 + 225. iT - 7.81e4T^{2} \)
7 \( 1 - 6.20T + 8.23e5T^{2} \)
11 \( 1 - 5.45e3T + 1.94e7T^{2} \)
13 \( 1 + 9.33e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.42e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.61e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.07e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.38e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.66e5iT - 2.75e10T^{2} \)
41 \( 1 + 4.15e5T + 1.94e11T^{2} \)
43 \( 1 - 9.00e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.85e5T + 5.06e11T^{2} \)
53 \( 1 - 1.40e6T + 1.17e12T^{2} \)
59 \( 1 + 1.03e6iT - 2.48e12T^{2} \)
61 \( 1 - 9.44e5iT - 3.14e12T^{2} \)
67 \( 1 - 1.87e6T + 6.06e12T^{2} \)
71 \( 1 - 3.16e6T + 9.09e12T^{2} \)
73 \( 1 + 5.00e6T + 1.10e13T^{2} \)
79 \( 1 - 4.38e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.16e6T + 2.71e13T^{2} \)
89 \( 1 - 9.98e6iT - 4.42e13T^{2} \)
97 \( 1 - 3.91e6iT - 8.07e13T^{2} \)
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

−14.80837211126916160006477123352, −13.36576317090839328550123511874, −11.91200455189252381118465752770, −11.29772368439599394769496513918, −9.557202510193599253915796069615, −8.160903955734266515062498786817, −6.59525440978438147286607331012, −5.19233921414730859074672056851, −2.95439760836915469285875884494, −0.925502751766746630100518064361, 1.87539648857833985703224770927, 3.57889000161704052356837233123, 6.08089433563285703215721524112, 6.91224210997182467507005605633, 8.653620933271445737837502767997, 10.43510460984823971292094647203, 11.38728167677347083486012948582, 12.29700186958769956411306709887, 14.46604873772855825859909348588, 14.66931696780961314512163921208

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