↑

Properties

Label 2-37-37.27-c5-0-13
Degree $2$
Conductor $37$
Sign $-0.999 + 0.0126i$
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  + (−6.08 − 3.51i)2-s + (3.15 + 5.46i)3-s + (8.66 + 15.0i)4-s + (30.9 − 17.8i)5-s − 44.3i·6-s + (−47.9 − 83.1i)7-s + 103. i·8-s + (101. − 175. i)9-s − 250.·10-s − 604.·11-s + (−54.6 + 94.7i)12-s + (−940. + 542. i)13-s + 673. i·14-s + (195. + 112. i)15-s + (639. − 1.10e3i)16-s + (−927. − 535. i)17-s + ⋯
L(s)  = 1  + (−1.07 − 0.620i)2-s + (0.202 + 0.350i)3-s + (0.270 + 0.468i)4-s + (0.553 − 0.319i)5-s − 0.502i·6-s + (−0.370 − 0.640i)7-s + 0.569i·8-s + (0.417 − 0.723i)9-s − 0.793·10-s − 1.50·11-s + (−0.109 + 0.189i)12-s + (−1.54 + 0.890i)13-s + 0.918i·14-s + (0.224 + 0.129i)15-s + (0.624 − 1.08i)16-s + (−0.778 − 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.999 + 0.0126i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.999 + 0.0126i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00242667 - 0.383101i\)
\(L(\frac12)\) \(\approx\) \(0.00242667 - 0.383101i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.79e3 - 8.13e3i)T \)
good2 \( 1 + (6.08 + 3.51i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (-3.15 - 5.46i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-30.9 + 17.8i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (47.9 + 83.1i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 604.T + 1.61e5T^{2} \)
13 \( 1 + (940. - 542. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (927. + 535. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-305. + 176. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 4.77e3iT - 6.43e6T^{2} \)
29 \( 1 - 646. iT - 2.05e7T^{2} \)
31 \( 1 - 105. iT - 2.86e7T^{2} \)
41 \( 1 + (6.89e3 + 1.19e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 3.81e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.50e4T + 2.29e8T^{2} \)
53 \( 1 + (-5.08e3 + 8.81e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.59e4 - 1.50e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-6.38e3 + 3.68e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.86e4 - 3.23e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (3.67e4 + 6.36e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 4.78e4T + 2.07e9T^{2} \)
79 \( 1 + (-7.90e4 + 4.56e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.70e4 + 2.95e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-1.66e3 - 959. i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.13e3iT - 8.58e9T^{2} \)
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

−14.82835593065580959812527910448, −13.48140869673810419388999475950, −12.13513034451944400049441929241, −10.48044979700164103617421697675, −9.835260175031314250651439933357, −8.821680579354266581575481082559, −7.13082874931297820527626406722, −4.85088024984423791767773582594, −2.41011314167421427327168080743, −0.28416284055065145227887872901, 2.42131524943366260987508694323, 5.51515953303276001335971863809, 7.25516735703199990158187130349, 8.077413438364361147208343595335, 9.638984203500458777514928320693, 10.44603487789349446427504689499, 12.61265137613697646887721997812, 13.46014446127004231281820768730, 15.25410869305828362908936397146, 15.94238175922921431886450648782

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