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Properties

Label 2-666-9.7-c1-0-35
Degree $2$
Conductor $666$
Sign $-0.285 - 0.958i$
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 + (−0.398 − 1.68i)3-s + (−0.499 − 0.866i)4-s + (−0.265 − 0.459i)5-s + (−1.65 − 0.497i)6-s + (−1.21 + 2.10i)7-s − 0.999·8-s + (−2.68 + 1.34i)9-s − 0.530·10-s + (−1.62 + 2.81i)11-s + (−1.26 + 1.18i)12-s + (−0.625 − 1.08i)13-s + (1.21 + 2.10i)14-s + (−0.668 + 0.630i)15-s + (−0.5 + 0.866i)16-s − 4.30·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.230 − 0.973i)3-s + (−0.249 − 0.433i)4-s + (−0.118 − 0.205i)5-s + (−0.677 − 0.203i)6-s + (−0.458 + 0.793i)7-s − 0.353·8-s + (−0.894 + 0.448i)9-s − 0.167·10-s + (−0.490 + 0.849i)11-s + (−0.363 + 0.342i)12-s + (−0.173 − 0.300i)13-s + (0.324 + 0.561i)14-s + (−0.172 + 0.162i)15-s + (−0.125 + 0.216i)16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0866004 + 0.116218i\)
\(L(\frac12)\) \(\approx\) \(0.0866004 + 0.116218i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.398 + 1.68i)T \)
37 \( 1 + T \)
good5 \( 1 + (0.265 + 0.459i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.21 - 2.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.62 - 2.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.625 + 1.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.30T + 17T^{2} \)
19 \( 1 + 6.96T + 19T^{2} \)
23 \( 1 + (2.88 + 5.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.31 - 2.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.11 - 8.86i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 + (3.80 + 6.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.07 + 5.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.65T + 53T^{2} \)
59 \( 1 + (-1.92 - 3.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.83 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.424 - 0.736i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.49T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + (6.50 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.220 + 0.382i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.81T + 89T^{2} \)
97 \( 1 + (3.23 - 5.60i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.26139931358078983043186484434, −8.771441287418223125698563197519, −8.472441472204626049666511561069, −7.00044358832037475447432284258, −6.36031427998455758034670845571, −5.29687691158105837655922114386, −4.35944369857014669272702401871, −2.69283261074849895380162808000, −2.03523335313844655746188314628, −0.06512307515541410523907962725, 2.83081837904375735867741914021, 3.97182405197889010659440978290, 4.54618168062127372985193945364, 5.85921445788580427663535748939, 6.43961871308282064478872807530, 7.58928450520148934810693592273, 8.530847481981040897150047800492, 9.402663214930067688783219614489, 10.28097737411190395136922735947, 11.08327977496313370339176178427

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