↑

Properties

Label 2-37-37.6-c6-0-10
Degree $2$
Conductor $37$
Sign $0.898 + 0.438i$
Analytic cond. $8.51200$
Root an. cond. $2.91753$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.77 − 2.77i)2-s − 27.0i·3-s + 48.5i·4-s + (96.7 + 96.7i)5-s + (−75.0 − 75.0i)6-s + 312.·7-s + (312. + 312. i)8-s − 0.728·9-s + 537.·10-s − 1.62e3i·11-s + 1.31e3·12-s + (−1.59e3 − 1.59e3i)13-s + (867. − 867. i)14-s + (2.61e3 − 2.61e3i)15-s − 1.36e3·16-s + (1.95e3 + 1.95e3i)17-s + ⋯
L(s)  = 1  + (0.347 − 0.347i)2-s − 1.00i·3-s + 0.758i·4-s + (0.774 + 0.774i)5-s + (−0.347 − 0.347i)6-s + 0.910·7-s + (0.610 + 0.610i)8-s − 0.000999·9-s + 0.537·10-s − 1.21i·11-s + 0.759·12-s + (−0.728 − 0.728i)13-s + (0.316 − 0.316i)14-s + (0.774 − 0.774i)15-s − 0.334·16-s + (0.398 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ 0.898 + 0.438i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.43844 - 0.563678i\)
\(L(\frac12)\) \(\approx\) \(2.43844 - 0.563678i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.03e4 + 4.63e4i)T \)
good2 \( 1 + (-2.77 + 2.77i)T - 64iT^{2} \)
3 \( 1 + 27.0iT - 729T^{2} \)
5 \( 1 + (-96.7 - 96.7i)T + 1.56e4iT^{2} \)
7 \( 1 - 312.T + 1.17e5T^{2} \)
11 \( 1 + 1.62e3iT - 1.77e6T^{2} \)
13 \( 1 + (1.59e3 + 1.59e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-1.95e3 - 1.95e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-9.30e3 - 9.30e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-3.52e3 - 3.52e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (-6.58e3 + 6.58e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (3.14e4 - 3.14e4i)T - 8.87e8iT^{2} \)
41 \( 1 - 1.01e3iT - 4.75e9T^{2} \)
43 \( 1 + (6.21e4 + 6.21e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 9.56e4T + 1.07e10T^{2} \)
53 \( 1 + 1.16e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.67e5 - 1.67e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (2.67e3 - 2.67e3i)T - 5.15e10iT^{2} \)
67 \( 1 - 4.09e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.96e5T + 1.28e11T^{2} \)
73 \( 1 + 5.40e5iT - 1.51e11T^{2} \)
79 \( 1 + (1.21e5 + 1.21e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 9.27e3T + 3.26e11T^{2} \)
89 \( 1 + (7.37e5 - 7.37e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (8.69e4 + 8.69e4i)T + 8.32e11iT^{2} \)
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

−14.41696937641026541142792883345, −13.77962146701316512701075985988, −12.63853902402572230074954768296, −11.62162880809116712228117014115, −10.33746049825680136615707191134, −8.210157830840850350058751688093, −7.27700863871277811459027225236, −5.56669619981043198012413429900, −3.20859753672963274215132296722, −1.66963603598952254972286430463, 1.57783405542670520153628648783, 4.80727007937080853690208457672, 5.00960041842108597207399219043, 7.13218648947105620869876897742, 9.410104374047580573218045784730, 9.818343723805532852236121170440, 11.37508786448195507123583235245, 13.07097488924543961565577861020, 14.31995585513248216547347000427, 15.09908607470102616639915794150

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