Properties

Label 2-37-37.6-c6-0-1
Degree $2$
Conductor $37$
Sign $-0.425 - 0.905i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.77 − 5.77i)2-s + 42.6i·3-s − 2.70i·4-s + (−53.2 − 53.2i)5-s + (246. + 246. i)6-s − 452.·7-s + (353. + 353. i)8-s − 1.09e3·9-s − 615.·10-s + 1.79e3i·11-s + 115.·12-s + (−2.65e3 − 2.65e3i)13-s + (−2.61e3 + 2.61e3i)14-s + (2.27e3 − 2.27e3i)15-s + 4.26e3·16-s + (6.58e3 + 6.58e3i)17-s + ⋯
L(s)  = 1  + (0.721 − 0.721i)2-s + 1.57i·3-s − 0.0422i·4-s + (−0.426 − 0.426i)5-s + (1.14 + 1.14i)6-s − 1.31·7-s + (0.691 + 0.691i)8-s − 1.49·9-s − 0.615·10-s + 1.34i·11-s + 0.0667·12-s + (−1.20 − 1.20i)13-s + (−0.951 + 0.951i)14-s + (0.673 − 0.673i)15-s + 1.04·16-s + (1.34 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.425 - 0.905i$
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: \((2,\ 37,\ (\ :3),\ -0.425 - 0.905i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.846619 + 1.33299i\)
\(L(\frac12)\) \(\approx\) \(0.846619 + 1.33299i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.32e4 - 4.89e4i)T \)
good2 \( 1 + (-5.77 + 5.77i)T - 64iT^{2} \)
3 \( 1 - 42.6iT - 729T^{2} \)
5 \( 1 + (53.2 + 53.2i)T + 1.56e4iT^{2} \)
7 \( 1 + 452.T + 1.17e5T^{2} \)
11 \( 1 - 1.79e3iT - 1.77e6T^{2} \)
13 \( 1 + (2.65e3 + 2.65e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-6.58e3 - 6.58e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-6.30e3 - 6.30e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (1.98e3 + 1.98e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (-1.93e4 + 1.93e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (1.51e4 - 1.51e4i)T - 8.87e8iT^{2} \)
41 \( 1 + 2.55e4iT - 4.75e9T^{2} \)
43 \( 1 + (-2.99e4 - 2.99e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 4.70e4T + 1.07e10T^{2} \)
53 \( 1 + 5.07e4T + 2.21e10T^{2} \)
59 \( 1 + (-4.15e4 - 4.15e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.22e5 + 1.22e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 1.65e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.22e5T + 1.28e11T^{2} \)
73 \( 1 + 3.02e5iT - 1.51e11T^{2} \)
79 \( 1 + (-1.73e5 - 1.73e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 6.35e5T + 3.26e11T^{2} \)
89 \( 1 + (5.66e5 - 5.66e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (4.92e5 + 4.92e5i)T + 8.32e11iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.43444033649061801687042456898, −14.44159831246744428791746453569, −12.57387090555163410645873236083, −12.23578363393213004334877916517, −10.18931726398359814078801117827, −9.957955851146258659345992173120, −7.939713526644527807928758090632, −5.32645006526789568938900581154, −4.10476896110670592701551930216, −3.07969614147522056733090053900, 0.62947811338519286598165141263, 3.09133445971963643511911959461, 5.59777645229370947185133951266, 6.90889805098007050464845865728, 7.33778537291058965147512334882, 9.505577803578640624466857727027, 11.54583419920476108540431468642, 12.62621766914804764891977791396, 13.77762428714252230680928018482, 14.23835956416779112943711961546

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