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Properties

Label 2-666-37.26-c1-0-2
Degree $2$
Conductor $666$
Sign $0.227 - 0.973i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1 + 1.73i)7-s + 0.999·8-s − 0.999·10-s − 2·11-s + (−3 + 5.19i)13-s + 1.99·14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s + (1 − 1.73i)19-s + (0.499 + 0.866i)20-s + (1 + 1.73i)22-s − 8·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + (−0.832 + 1.44i)13-s + 0.534·14-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s + (0.111 + 0.193i)20-s + (0.213 + 0.369i)22-s − 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.227 - 0.973i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543113 + 0.430903i\)
\(L(\frac12)\) \(\approx\) \(0.543113 + 0.430903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
37 \( 1 + (5.5 - 2.59i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−10.57558942196640710174124815119, −9.749414948887695676405478890478, −9.175968016865979394587002543034, −8.319472851834525247541570337237, −7.33572303700705219607863314739, −6.20885108567918048370892269039, −5.14422654700363670485629718103, −4.11469300728018895072364159528, −2.76565167002142138433502680168, −1.72851462967080437276638280968, 0.40129317819854791914808432228, 2.45697887136000946972439680267, 3.72187421079470915592075685526, 5.11325925422841372841486679924, 5.85917460319188376724296717262, 6.96233728797908714144420624898, 7.65950797607866890725037612800, 8.357015427386739172283851771363, 9.724100544094617858624987398541, 10.16356137322599288057736205662

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