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Properties

Label 2-666-37.33-c1-0-8
Degree $2$
Conductor $666$
Sign $0.482 + 0.876i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.491 + 0.412i)5-s + (−1.28 + 1.08i)7-s + (−0.5 + 0.866i)8-s + (0.320 + 0.555i)10-s + (1.23 − 2.14i)11-s + (6.53 + 2.37i)13-s + (0.840 + 1.45i)14-s + (0.766 + 0.642i)16-s + (4.70 − 1.71i)17-s + (−0.902 − 5.11i)19-s + (0.602 − 0.219i)20-s + (−1.89 − 1.58i)22-s + (−2.86 − 4.95i)23-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.469 − 0.171i)4-s + (−0.219 + 0.184i)5-s + (−0.486 + 0.408i)7-s + (−0.176 + 0.306i)8-s + (0.101 + 0.175i)10-s + (0.372 − 0.645i)11-s + (1.81 + 0.659i)13-s + (0.224 + 0.389i)14-s + (0.191 + 0.160i)16-s + (1.14 − 0.415i)17-s + (−0.206 − 1.17i)19-s + (0.134 − 0.0490i)20-s + (−0.403 − 0.338i)22-s + (−0.596 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.482 + 0.876i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.482 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29644 - 0.766366i\)
\(L(\frac12)\) \(\approx\) \(1.29644 - 0.766366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 \)
37 \( 1 + (-2.16 - 5.68i)T \)
good5 \( 1 + (0.491 - 0.412i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.28 - 1.08i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.53 - 2.37i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.70 + 1.71i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.902 + 5.11i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (2.86 + 4.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.20 + 9.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
41 \( 1 + (3.16 + 1.15i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + (-5.07 - 8.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 + 4.43i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-9.62 - 8.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.345 + 0.125i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-9.53 + 8.00i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.59 + 14.6i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 4.52T + 73T^{2} \)
79 \( 1 + (1.54 - 1.29i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (10.0 - 3.65i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-0.827 - 0.694i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.0254 - 0.0440i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.45886183783250222102685291659, −9.563299771870126609239332494738, −8.759840302216141078140388477761, −8.035999775757215838434885876084, −6.50708125209455387473969398600, −6.01064007639282518303773378914, −4.59964094636042699843345533112, −3.59498389448925893856081614078, −2.69515855776599726235494489052, −1.01063509347770562424194331876, 1.25103253613322918980937804074, 3.45469625108080685549166813470, 4.02217091429304649284031400588, 5.44342230490501707238007105950, 6.18686239652212831250656128131, 7.08005383888510302662582756077, 8.158377111412883343085423171025, 8.571995780967154979077082372325, 9.965512972210817639837232109635, 10.31445780703931473996588015401

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