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Properties

Label 2-74-37.3-c1-0-1
Degree $2$
Conductor $74$
Sign $0.717 + 0.696i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 − 0.766i)2-s + (1.43 − 1.20i)3-s + (−0.173 − 0.984i)4-s + (−1.45 + 4.00i)5-s − 1.87i·6-s + (−3.39 − 1.23i)7-s + (−0.866 − 0.500i)8-s + (0.0923 − 0.524i)9-s + (2.13 + 3.69i)10-s + (1.05 − 1.83i)11-s + (−1.43 − 1.20i)12-s + (2.84 − 0.500i)13-s + (−3.12 + 1.80i)14-s + (2.74 + 7.53i)15-s + (−0.939 + 0.342i)16-s + (0.0263 + 0.00463i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.831 − 0.697i)3-s + (−0.0868 − 0.492i)4-s + (−0.652 + 1.79i)5-s − 0.767i·6-s + (−1.28 − 0.466i)7-s + (−0.306 − 0.176i)8-s + (0.0307 − 0.174i)9-s + (0.674 + 1.16i)10-s + (0.319 − 0.552i)11-s + (−0.415 − 0.348i)12-s + (0.787 − 0.138i)13-s + (−0.835 + 0.482i)14-s + (0.707 + 1.94i)15-s + (−0.234 + 0.0855i)16-s + (0.00638 + 0.00112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11708 - 0.452583i\)
\(L(\frac12)\) \(\approx\) \(1.11708 - 0.452583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
37 \( 1 + (4.49 + 4.09i)T \)
good3 \( 1 + (-1.43 + 1.20i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (1.45 - 4.00i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (3.39 + 1.23i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.84 + 0.500i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.0263 - 0.00463i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (2.07 + 2.47i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.57 + 1.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.96 - 2.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.76iT - 31T^{2} \)
41 \( 1 + (-0.259 - 1.46i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 5.53iT - 43T^{2} \)
47 \( 1 + (1.30 + 2.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.79 - 0.652i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.92 - 5.28i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.65 - 0.996i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.50 + 2.36i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.10 - 5.96i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 + (0.484 - 1.32i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.294 - 1.67i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.82 - 7.77i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.65 + 3.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.11011426200744505690565111232, −13.59858349823213489218579303670, −12.44407441174347952798640188068, −11.01709147654437644300796222586, −10.37943377493979445805594529191, −8.687196328610354018412793528136, −7.12413297579495938977416186534, −6.46696189581166297350921686575, −3.57366065209058556957467984464, −2.85748680373103714221288870135, 3.56608600280887833588013475599, 4.57894080905580908287929945523, 6.18731652692186475958159454041, 8.090235817728825550914116417678, 8.975042040887139885815403727142, 9.650959651129666869425297339778, 11.92303541377000272756020418805, 12.71532386091061283943739215057, 13.56014237543109086876126528752, 15.05683259438011521656808106087

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