Properties

Label 2-37-37.10-c7-0-20
Degree $2$
Conductor $37$
Sign $-0.473 - 0.880i$
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  + (7.43 − 12.8i)2-s + (−34.7 − 60.2i)3-s + (−46.6 − 80.7i)4-s + (−238. − 412. i)5-s − 1.03e3·6-s + (249. + 431. i)7-s + 516.·8-s + (−1.32e3 + 2.29e3i)9-s − 7.08e3·10-s + 7.32e3·11-s + (−3.24e3 + 5.62e3i)12-s + (−5.10e3 − 8.83e3i)13-s + 7.41e3·14-s + (−1.65e4 + 2.87e4i)15-s + (9.81e3 − 1.69e4i)16-s + (−1.40e4 + 2.43e4i)17-s + ⋯
L(s)  = 1  + (0.657 − 1.13i)2-s + (−0.743 − 1.28i)3-s + (−0.364 − 0.631i)4-s + (−0.852 − 1.47i)5-s − 1.95·6-s + (0.274 + 0.475i)7-s + 0.356·8-s + (−0.607 + 1.05i)9-s − 2.24·10-s + 1.66·11-s + (−0.542 + 0.939i)12-s + (−0.644 − 1.11i)13-s + 0.722·14-s + (−1.26 + 2.19i)15-s + (0.598 − 1.03i)16-s + (−0.692 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.808159 + 1.35266i\)
\(L(\frac12)\) \(\approx\) \(0.808159 + 1.35266i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.98e5 + 2.35e5i)T \)
good2 \( 1 + (-7.43 + 12.8i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (34.7 + 60.2i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (238. + 412. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-249. - 431. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 7.32e3T + 1.94e7T^{2} \)
13 \( 1 + (5.10e3 + 8.83e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (1.40e4 - 2.43e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.08e3 + 3.61e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 6.04e4T + 3.40e9T^{2} \)
29 \( 1 + 1.65e5T + 1.72e10T^{2} \)
31 \( 1 + 1.91e4T + 2.75e10T^{2} \)
41 \( 1 + (2.53e5 + 4.38e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 1.37e5T + 2.71e11T^{2} \)
47 \( 1 - 5.98e5T + 5.06e11T^{2} \)
53 \( 1 + (3.44e5 - 5.97e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (3.84e5 - 6.66e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-6.06e5 - 1.05e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.41e6 - 2.45e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (1.28e6 + 2.22e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 1.54e6T + 1.10e13T^{2} \)
79 \( 1 + (2.85e6 + 4.95e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.55e5 + 2.68e5i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-1.03e6 + 1.79e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 7.63e6T + 8.07e13T^{2} \)
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

−13.19111678495925475761096091309, −12.55807134640077109823026885753, −11.99891438661384520632183990433, −11.13299533511015988197191470559, −8.861134604874756755839214781970, −7.43644840020992368846220007267, −5.51363954742790741873300410215, −4.08382365934795560890520817458, −1.71485780994007168255614004229, −0.66357531979768961037769658234, 3.83831350393547413894768065077, 4.68373093920933451224451335216, 6.51562947283769647213376664514, 7.22964429312596513717766713307, 9.530393965049458854005551070219, 11.05459195937778741628033243089, 11.53161969007454787946611296968, 14.06697040033058518231337483486, 14.74226429472683275355553724307, 15.45357877417486596206300062289

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