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Properties

Label 2-37-37.6-c6-0-4
Degree $2$
Conductor $37$
Sign $-0.941 - 0.337i$
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  + (−0.186 + 0.186i)2-s + 44.0i·3-s + 63.9i·4-s + (56.4 + 56.4i)5-s + (−8.18 − 8.18i)6-s + 89.8·7-s + (−23.8 − 23.8i)8-s − 1.20e3·9-s − 21.0·10-s − 1.85e3i·11-s − 2.81e3·12-s + (2.22e3 + 2.22e3i)13-s + (−16.7 + 16.7i)14-s + (−2.48e3 + 2.48e3i)15-s − 4.08e3·16-s + (640. + 640. i)17-s + ⋯
L(s)  = 1  + (−0.0232 + 0.0232i)2-s + 1.63i·3-s + 0.998i·4-s + (0.451 + 0.451i)5-s + (−0.0379 − 0.0379i)6-s + 0.261·7-s + (−0.0464 − 0.0464i)8-s − 1.65·9-s − 0.0210·10-s − 1.39i·11-s − 1.62·12-s + (1.01 + 1.01i)13-s + (−0.00609 + 0.00609i)14-s + (−0.736 + 0.736i)15-s − 0.996·16-s + (0.130 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.941 - 0.337i$
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: \((2,\ 37,\ (\ :3),\ -0.941 - 0.337i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.288710 + 1.66072i\)
\(L(\frac12)\) \(\approx\) \(0.288710 + 1.66072i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.53e4 - 4.38e4i)T \)
good2 \( 1 + (0.186 - 0.186i)T - 64iT^{2} \)
3 \( 1 - 44.0iT - 729T^{2} \)
5 \( 1 + (-56.4 - 56.4i)T + 1.56e4iT^{2} \)
7 \( 1 - 89.8T + 1.17e5T^{2} \)
11 \( 1 + 1.85e3iT - 1.77e6T^{2} \)
13 \( 1 + (-2.22e3 - 2.22e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-640. - 640. i)T + 2.41e7iT^{2} \)
19 \( 1 + (-636. - 636. i)T + 4.70e7iT^{2} \)
23 \( 1 + (-472. - 472. i)T + 1.48e8iT^{2} \)
29 \( 1 + (4.74e3 - 4.74e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (-1.06e4 + 1.06e4i)T - 8.87e8iT^{2} \)
41 \( 1 + 4.87e4iT - 4.75e9T^{2} \)
43 \( 1 + (-1.03e5 - 1.03e5i)T + 6.32e9iT^{2} \)
47 \( 1 - 505.T + 1.07e10T^{2} \)
53 \( 1 + 2.04e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.72e5 - 1.72e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.23e5 - 1.23e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 4.98e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.69e5T + 1.28e11T^{2} \)
73 \( 1 + 4.18e5iT - 1.51e11T^{2} \)
79 \( 1 + (4.45e5 + 4.45e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 8.96e5T + 3.26e11T^{2} \)
89 \( 1 + (-9.65e5 + 9.65e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (3.81e5 + 3.81e5i)T + 8.32e11iT^{2} \)
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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

−16.03415004277343657814591546095, −14.49646072169849435688560044937, −13.53272162432293550382346030404, −11.55612126469341564129470870116, −10.78630052593847146668376835257, −9.321389602358681527107873042672, −8.328920238189814329142611997713, −6.13692598448220894733294234464, −4.28604748229198077248463163116, −3.12583524524165421711221958558, 0.913263544809134488436069782248, 1.98654986659142922044586199570, 5.30898476318250591231271460765, 6.52596558604273659742566798460, 7.87148107979235981714320570895, 9.438262971826947699900300870569, 10.97889539229209662658420856568, 12.46027640173297547394864295707, 13.27751141123479074829677833145, 14.31383370190372255985681615704

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