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Properties

Label 2-666-37.36-c1-0-3
Degree $2$
Conductor $666$
Sign $0.753 - 0.657i$
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  − i·2-s − 4-s + 0.791i·5-s − 2·7-s + i·8-s + 0.791·10-s + 0.791·11-s + 3.79i·13-s + 2i·14-s + 16-s + 7.58i·17-s − 1.58i·19-s − 0.791i·20-s − 0.791i·22-s − 3.79i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.353i·5-s − 0.755·7-s + 0.353i·8-s + 0.250·10-s + 0.238·11-s + 1.05i·13-s + 0.534i·14-s + 0.250·16-s + 1.83i·17-s − 0.363i·19-s − 0.176i·20-s − 0.168i·22-s − 0.790i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.997529 + 0.374120i\)
\(L(\frac12)\) \(\approx\) \(0.997529 + 0.374120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
37 \( 1 + (-4 - 4.58i)T \)
good5 \( 1 - 0.791iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 - 3.79iT - 13T^{2} \)
17 \( 1 - 7.58iT - 17T^{2} \)
19 \( 1 + 1.58iT - 19T^{2} \)
23 \( 1 + 3.79iT - 23T^{2} \)
29 \( 1 - 3.79iT - 29T^{2} \)
31 \( 1 - 8.37iT - 31T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 - 1.58T + 53T^{2} \)
59 \( 1 - 1.58iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
79 \( 1 + 8.20iT - 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.61713883420870395317079493416, −9.932453579024374376980950710534, −8.962266431670756372909187898364, −8.323466951552818569124632541823, −6.82215596125987250932608737995, −6.36071353204502605959957554232, −4.91661028018966876415874878921, −3.86779797057068268703741309786, −2.93923572956659985808711792053, −1.56949067755688851094860591461, 0.59634627952344899064702322491, 2.77463570315436474673383872540, 3.94679880411033682213202880664, 5.16885103393077529087721377706, 5.86193379646288735701446779589, 6.97347293477232044839382933831, 7.65635891321490626260072204066, 8.675166967590396239907994598174, 9.522007494150498095193635572829, 10.07568695347310196161620121973

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