Properties

Label 2-74-37.16-c1-0-2
Degree $2$
Conductor $74$
Sign $0.936 - 0.349i$
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  + (0.939 + 0.342i)2-s + (−0.326 + 0.118i)3-s + (0.766 + 0.642i)4-s + (−0.0209 + 0.118i)5-s − 0.347·6-s + (0.233 − 1.32i)7-s + (0.500 + 0.866i)8-s + (−2.20 + 1.85i)9-s + (−0.0603 + 0.104i)10-s + (−2.26 − 3.92i)11-s + (−0.326 − 0.118i)12-s + (−0.592 − 0.497i)13-s + (0.673 − 1.16i)14-s + (−0.00727 − 0.0412i)15-s + (0.173 + 0.984i)16-s + (−2.29 + 1.92i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.188 + 0.0685i)3-s + (0.383 + 0.321i)4-s + (−0.00936 + 0.0531i)5-s − 0.141·6-s + (0.0884 − 0.501i)7-s + (0.176 + 0.306i)8-s + (−0.735 + 0.616i)9-s + (−0.0190 + 0.0330i)10-s + (−0.683 − 1.18i)11-s + (−0.0942 − 0.0342i)12-s + (−0.164 − 0.137i)13-s + (0.180 − 0.311i)14-s + (−0.00187 − 0.0106i)15-s + (0.0434 + 0.246i)16-s + (−0.557 + 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15734 + 0.209014i\)
\(L(\frac12)\) \(\approx\) \(1.15734 + 0.209014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-5.60 - 2.36i)T \)
good3 \( 1 + (0.326 - 0.118i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.0209 - 0.118i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.233 + 1.32i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.26 + 3.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.592 + 0.497i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.29 - 1.92i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-1.91 + 0.698i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.0282 - 0.0488i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.89 - 5.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.34T + 31T^{2} \)
41 \( 1 + (5.47 + 4.59i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 9.31T + 43T^{2} \)
47 \( 1 + (4.25 - 7.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.482 - 2.73i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.25 + 12.7i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.82 + 7.40i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.889 + 5.04i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.6 - 4.59i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 8.71T + 73T^{2} \)
79 \( 1 + (-0.720 + 4.08i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.53 - 3.80i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-1.32 - 7.50i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-5.12 + 8.86i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.42281053767039294682612504069, −13.68284624814304050455128357737, −12.69724059537500203909278302141, −11.21179795472905936229026279947, −10.66353147284908175165575272152, −8.710383885911537810338729607655, −7.55697696988713929644103738205, −6.04520785006568581864829579046, −4.88238000245229935233914848553, −3.06339778718228971833146906353, 2.63843301892992575746397326465, 4.61646962993621454086158432368, 5.88364167419000677108053926181, 7.27587729921328182570226523607, 8.934005847936948491122467036946, 10.20451060563796322255856316875, 11.56276664294185113003519060371, 12.27887207316983985713756700163, 13.32480792506424814608856192091, 14.61025949884226299351068645467

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