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Properties

Label 2-666-37.25-c1-0-2
Degree $2$
Conductor $666$
Sign $-0.560 - 0.827i$
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  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (1.45 + 4.00i)5-s + (−3.39 + 1.23i)7-s + (0.866 − 0.500i)8-s + (2.13 − 3.69i)10-s + (−1.05 − 1.83i)11-s + (2.84 + 0.500i)13-s + (3.12 + 1.80i)14-s + (−0.939 − 0.342i)16-s + (−0.0263 + 0.00463i)17-s + (−2.07 + 2.47i)19-s + (−4.19 + 0.740i)20-s + (−0.723 + 1.98i)22-s + (−2.57 − 1.48i)23-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.0868 + 0.492i)4-s + (0.652 + 1.79i)5-s + (−1.28 + 0.466i)7-s + (0.306 − 0.176i)8-s + (0.674 − 1.16i)10-s + (−0.319 − 0.552i)11-s + (0.787 + 0.138i)13-s + (0.835 + 0.482i)14-s + (−0.234 − 0.0855i)16-s + (−0.00638 + 0.00112i)17-s + (−0.476 + 0.567i)19-s + (−0.939 + 0.165i)20-s + (−0.154 + 0.424i)22-s + (−0.536 − 0.309i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.560 - 0.827i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.560 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334155 + 0.629919i\)
\(L(\frac12)\) \(\approx\) \(0.334155 + 0.629919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 \)
37 \( 1 + (4.49 - 4.09i)T \)
good5 \( 1 + (-1.45 - 4.00i)T + (-3.83 + 3.21i)T^{2} \)
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

−10.75115103989673416862271449923, −9.990302545629250524351918911202, −9.433838584064814844468620603587, −8.336481860913834700665925532606, −7.21842877009367913748146555556, −6.30111497617667418558101750256, −5.87210641159737037369707634644, −3.71889521796337368214768978177, −3.05751545129879217897027002828, −2.11115873107271349391616793604, 0.42104885896602155377247312217, 1.85613112869709213880136028256, 3.79021518191474511792319508309, 4.90657956079575490120439221821, 5.76705483489460115931633467096, 6.58900583767742178081651760621, 7.68860433655589297615717081021, 8.699728900139901281077897574783, 9.242395947510552465178049206677, 9.921084997584740592968970922805

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