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Properties

Label 2-666-37.34-c1-0-6
Degree $2$
Conductor $666$
Sign $0.939 - 0.342i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (1.43 + 0.524i)5-s + (4.53 + 1.65i)7-s + (0.500 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.546 + 0.947i)11-s + (0.307 + 1.74i)13-s + (−2.41 − 4.17i)14-s + (−0.939 + 0.342i)16-s + (−0.511 + 2.90i)17-s + (−0.632 + 0.530i)19-s + (−0.266 + 1.50i)20-s + (1.02 − 0.374i)22-s + (−0.121 − 0.210i)23-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.643 + 0.234i)5-s + (1.71 + 0.623i)7-s + (0.176 − 0.306i)8-s + (−0.242 − 0.419i)10-s + (−0.164 + 0.285i)11-s + (0.0852 + 0.483i)13-s + (−0.644 − 1.11i)14-s + (−0.234 + 0.0855i)16-s + (−0.124 + 0.703i)17-s + (−0.145 + 0.121i)19-s + (−0.0594 + 0.337i)20-s + (0.219 − 0.0797i)22-s + (−0.0253 − 0.0439i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45615 + 0.256917i\)
\(L(\frac12)\) \(\approx\) \(1.45615 + 0.256917i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 \)
37 \( 1 + (-5.84 + 1.69i)T \)
good5 \( 1 + (-1.43 - 0.524i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-4.53 - 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.546 - 0.947i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.307 - 1.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.511 - 2.90i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (0.632 - 0.530i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (0.121 + 0.210i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.78 - 4.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
41 \( 1 + (0.505 + 2.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 4.37T + 43T^{2} \)
47 \( 1 + (-1.13 - 1.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.77 + 2.10i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-8.29 + 3.01i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.04 + 11.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.44 + 3.07i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.933 + 0.783i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 8.79T + 73T^{2} \)
79 \( 1 + (2.02 + 0.736i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.884 - 5.01i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-8.43 + 3.07i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.17 + 3.77i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.76254356678216262070930916234, −9.670697032863699517995591067385, −8.868007695149237431139297198649, −8.136711424274345740947142006291, −7.28908498799380580009480356314, −6.02631592875668776271723636111, −5.08745135309341590107914954800, −3.97979830619848719393943985553, −2.29752638458697536135342307864, −1.66935969382836991639068339853, 1.06027775630635810880656907258, 2.27138934187053137667970948432, 4.16834613113443721761763960802, 5.20027434447101475535605374019, 5.85960998830782842372634104047, 7.24783973275895608451230724388, 7.82005561681991849639483707229, 8.656023980901936301586544020838, 9.518931418300034397856254472672, 10.44929782501590994735348341309

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