↑

Properties

Label 2-37-37.36-c5-0-14
Degree $2$
Conductor $37$
Sign $-0.938 - 0.344i$
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  − 8.11i·2-s + 2.07·3-s − 33.8·4-s − 43.8i·5-s − 16.8i·6-s − 152.·7-s + 15.1i·8-s − 238.·9-s − 356.·10-s + 531.·11-s − 70.1·12-s + 638. i·13-s + 1.23e3i·14-s − 90.8i·15-s − 960.·16-s − 1.62e3i·17-s + ⋯
L(s)  = 1  − 1.43i·2-s + 0.132·3-s − 1.05·4-s − 0.784i·5-s − 0.190i·6-s − 1.17·7-s + 0.0834i·8-s − 0.982·9-s − 1.12·10-s + 1.32·11-s − 0.140·12-s + 1.04i·13-s + 1.68i·14-s − 0.104i·15-s − 0.938·16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.938 - 0.344i$
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: \((2,\ 37,\ (\ :5/2),\ -0.938 - 0.344i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.194963 + 1.09858i\)
\(L(\frac12)\) \(\approx\) \(0.194963 + 1.09858i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-7.81e3 - 2.86e3i)T \)
good2 \( 1 + 8.11iT - 32T^{2} \)
3 \( 1 - 2.07T + 243T^{2} \)
5 \( 1 + 43.8iT - 3.12e3T^{2} \)
7 \( 1 + 152.T + 1.68e4T^{2} \)
11 \( 1 - 531.T + 1.61e5T^{2} \)
13 \( 1 - 638. iT - 3.71e5T^{2} \)
17 \( 1 + 1.62e3iT - 1.41e6T^{2} \)
19 \( 1 + 860. iT - 2.47e6T^{2} \)
23 \( 1 + 2.50e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.32e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.84e3iT - 2.86e7T^{2} \)
41 \( 1 - 1.62e4T + 1.15e8T^{2} \)
43 \( 1 + 6.23e3iT - 1.47e8T^{2} \)
47 \( 1 + 6.21e3T + 2.29e8T^{2} \)
53 \( 1 + 2.25e4T + 4.18e8T^{2} \)
59 \( 1 - 4.97e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.25e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.25e4T + 1.35e9T^{2} \)
71 \( 1 - 2.13e4T + 1.80e9T^{2} \)
73 \( 1 - 6.55e3T + 2.07e9T^{2} \)
79 \( 1 - 1.78e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.96e4T + 3.93e9T^{2} \)
89 \( 1 + 1.56e4iT - 5.58e9T^{2} \)
97 \( 1 + 9.22e4iT - 8.58e9T^{2} \)
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

−14.24873608973329304646648991261, −13.17014908234023586004669844768, −12.06434454513678106371432969105, −11.31406625501588419517579503860, −9.485376014742969800049546009268, −9.073114249170328034922893793565, −6.54076171170687543621527054585, −4.28269725195234123487878800794, −2.70344869477531619825049632818, −0.63462846189544086421684084853, 3.35952103149142659035498983904, 5.87254100437100845283545653290, 6.62386983829272303161051665201, 8.074094759335887928459722181480, 9.394382317747376631543942547472, 11.02171822532188273085206960974, 12.75135080315233043351779922842, 14.27224955589829091893527278939, 14.79710953570967420380888835705, 15.96358889721582263139312256375

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