Properties

Label 2-74-37.28-c1-0-1
Degree $2$
Conductor $74$
Sign $0.942 - 0.334i$
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  + (0.984 − 0.173i)2-s + (−0.266 + 1.50i)3-s + (0.939 − 0.342i)4-s + (−0.247 + 0.294i)5-s + 1.53i·6-s + (−2.50 − 2.09i)7-s + (0.866 − 0.5i)8-s + (0.613 + 0.223i)9-s + (−0.192 + 0.333i)10-s + (−1.29 − 2.23i)11-s + (0.266 + 1.50i)12-s + (−0.466 − 1.28i)13-s + (−2.82 − 1.63i)14-s + (−0.378 − 0.451i)15-s + (0.766 − 0.642i)16-s + (−1.13 + 3.13i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.153 + 0.871i)3-s + (0.469 − 0.171i)4-s + (−0.110 + 0.131i)5-s + 0.625i·6-s + (−0.945 − 0.793i)7-s + (0.306 − 0.176i)8-s + (0.204 + 0.0744i)9-s + (−0.0608 + 0.105i)10-s + (−0.389 − 0.674i)11-s + (0.0768 + 0.435i)12-s + (−0.129 − 0.355i)13-s + (−0.755 − 0.436i)14-s + (−0.0978 − 0.116i)15-s + (0.191 − 0.160i)16-s + (−0.276 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16844 + 0.201005i\)
\(L(\frac12)\) \(\approx\) \(1.16844 + 0.201005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
37 \( 1 + (-0.543 + 6.05i)T \)
good3 \( 1 + (0.266 - 1.50i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (0.247 - 0.294i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (2.50 + 2.09i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (1.29 + 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.466 + 1.28i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.13 - 3.13i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (5.89 + 1.03i)T + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (-6.53 - 3.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.78 + 1.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.53iT - 31T^{2} \)
41 \( 1 + (-7.77 + 2.82i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 - 4.33iT - 43T^{2} \)
47 \( 1 + (2.61 - 4.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.64 - 5.57i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-8.33 - 9.93i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.39 + 6.58i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.60 + 7.22i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.60 + 14.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + (-0.940 + 1.12i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.0104 - 0.00380i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (0.612 + 0.730i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-12.8 - 7.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.81391206304788343329366694347, −13.30756476310710664278851592650, −12.85443667651532735326832607887, −10.98799460386923159429163749379, −10.55725741757128463174548186183, −9.270724970515713181181972652884, −7.40711346034533821485602642102, −6.03380888885361611748833575069, −4.49469237045594641533512962211, −3.32864567113728287034536323364, 2.54674771238042052571562338493, 4.65317876991440684369376646399, 6.33084835150046139026160270806, 7.04874811397570534424476070680, 8.650453428374130119843433836975, 10.13053216161268090988534933826, 11.72657930687632417779776821824, 12.74123999058868198354530428339, 12.95776550460385992062539728262, 14.47797937456682821505776411752

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