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Properties

Label 2-666-111.8-c1-0-7
Degree $2$
Conductor $666$
Sign $0.600 + 0.799i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.205 + 0.765i)5-s + (−0.103 + 0.179i)7-s + (−0.707 + 0.707i)8-s + 0.792·10-s + 2.15·11-s + (3.34 − 0.896i)13-s + (0.146 + 0.146i)14-s + (0.500 + 0.866i)16-s + (−1.47 − 0.394i)17-s + (5.53 − 1.48i)19-s + (0.205 − 0.765i)20-s + (0.557 − 2.08i)22-s + (−0.186 + 0.186i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.0917 + 0.342i)5-s + (−0.0392 + 0.0679i)7-s + (−0.249 + 0.249i)8-s + 0.250·10-s + 0.649·11-s + (0.927 − 0.248i)13-s + (0.0392 + 0.0392i)14-s + (0.125 + 0.216i)16-s + (−0.357 − 0.0956i)17-s + (1.26 − 0.339i)19-s + (0.0458 − 0.171i)20-s + (0.118 − 0.443i)22-s + (−0.0388 + 0.0388i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52901 - 0.763744i\)
\(L(\frac12)\) \(\approx\) \(1.52901 - 0.763744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
37 \( 1 + (-5.85 + 1.66i)T \)
good5 \( 1 + (-0.205 - 0.765i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.103 - 0.179i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.47 + 0.394i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.53 + 1.48i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.186 - 0.186i)T - 23iT^{2} \)
29 \( 1 + (0.667 + 0.667i)T + 29iT^{2} \)
31 \( 1 + (-2.39 + 2.39i)T - 31iT^{2} \)
41 \( 1 + (2.63 - 4.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 + 1.86i)T + 43iT^{2} \)
47 \( 1 + 3.73iT - 47T^{2} \)
53 \( 1 + (0.970 - 0.560i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.83 + 1.83i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.320 + 1.19i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (6.44 + 3.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.47 - 2.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.83iT - 73T^{2} \)
79 \( 1 + (3.34 - 0.896i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.29 + 4.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.507 - 1.89i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.15 + 4.15i)T + 97iT^{2} \)
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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

−10.51251576070109077751412253749, −9.593461981277997341309709174772, −8.898259300839418803708681134807, −7.86891023372858099666431816283, −6.70409427741255474768371399092, −5.85917624308167510201974984698, −4.70725113253542381451038618250, −3.63280086091979688156531299912, −2.65940842112393880189001343408, −1.13925233186444972260261100897, 1.29569023629433021586727680618, 3.22746434661407429625462655682, 4.25538050247770726400347790500, 5.27976674613886918035892301856, 6.21575835649511817290087430557, 7.01515078120708275542238517004, 8.018376133746006848659885044394, 8.891885225729608108454327361178, 9.493393003804710855312006449517, 10.62002906821051631105634173954

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