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Properties

Label 2-37-37.31-c2-0-3
Degree $2$
Conductor $37$
Sign $0.810 - 0.585i$
Analytic cond. $1.00817$
Root an. cond. $1.00408$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.14 + 2.14i)2-s − 3.31i·3-s + 5.18i·4-s + (−1.68 + 1.68i)5-s + (7.09 − 7.09i)6-s − 11.3·7-s + (−2.53 + 2.53i)8-s − 1.95·9-s − 7.22·10-s + 8.99i·11-s + 17.1·12-s + (11.2 − 11.2i)13-s + (−24.3 − 24.3i)14-s + (5.58 + 5.58i)15-s + 9.86·16-s + (13.6 − 13.6i)17-s + ⋯
L(s)  = 1  + (1.07 + 1.07i)2-s − 1.10i·3-s + 1.29i·4-s + (−0.337 + 0.337i)5-s + (1.18 − 1.18i)6-s − 1.62·7-s + (−0.316 + 0.316i)8-s − 0.217·9-s − 0.722·10-s + 0.817i·11-s + 1.42·12-s + (0.862 − 0.862i)13-s + (−1.74 − 1.74i)14-s + (0.372 + 0.372i)15-s + 0.616·16-s + (0.801 − 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(1.00817\)
Root analytic conductor: \(1.00408\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1),\ 0.810 - 0.585i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43662 + 0.464272i\)
\(L(\frac12)\) \(\approx\) \(1.43662 + 0.464272i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (8.89 - 35.9i)T \)
good2 \( 1 + (-2.14 - 2.14i)T + 4iT^{2} \)
3 \( 1 + 3.31iT - 9T^{2} \)
5 \( 1 + (1.68 - 1.68i)T - 25iT^{2} \)
7 \( 1 + 11.3T + 49T^{2} \)
11 \( 1 - 8.99iT - 121T^{2} \)
13 \( 1 + (-11.2 + 11.2i)T - 169iT^{2} \)
17 \( 1 + (-13.6 + 13.6i)T - 289iT^{2} \)
19 \( 1 + (20.3 - 20.3i)T - 361iT^{2} \)
23 \( 1 + (4.76 - 4.76i)T - 529iT^{2} \)
29 \( 1 + (23.4 + 23.4i)T + 841iT^{2} \)
31 \( 1 + (-7.93 - 7.93i)T + 961iT^{2} \)
41 \( 1 + 48.4iT - 1.68e3T^{2} \)
43 \( 1 + (-15.1 + 15.1i)T - 1.84e3iT^{2} \)
47 \( 1 + 12.6T + 2.20e3T^{2} \)
53 \( 1 + 30.3T + 2.80e3T^{2} \)
59 \( 1 + (-17.1 + 17.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (8.21 + 8.21i)T + 3.72e3iT^{2} \)
67 \( 1 + 90.5iT - 4.48e3T^{2} \)
71 \( 1 + 48.8T + 5.04e3T^{2} \)
73 \( 1 - 95.5iT - 5.32e3T^{2} \)
79 \( 1 + (54.0 - 54.0i)T - 6.24e3iT^{2} \)
83 \( 1 - 108.T + 6.88e3T^{2} \)
89 \( 1 + (-39.5 - 39.5i)T + 7.92e3iT^{2} \)
97 \( 1 + (52.1 - 52.1i)T - 9.40e3iT^{2} \)
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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

−15.93756929462047643580555245960, −15.17427737950041112932403608480, −13.76451161859797224997977994822, −12.91526476400154075624666524862, −12.30076034277243912592778097812, −10.04371113016617227237573384958, −7.79514959343397166259272326476, −6.85413117995919767864037208342, −5.90382030488663769854483882434, −3.58542025964866390961872243458, 3.36705816911608493574357440622, 4.29883622849917802003891067842, 6.11254517303102304633034296900, 8.947396452317435931809089105762, 10.22553289736087342916674273034, 11.16639682364334538215750612848, 12.58233406042260158648540584630, 13.34912927400628459590132769987, 14.73864277073161146319288564475, 16.06072762766520922761868756377

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