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Properties

Label 2-666-111.8-c1-0-8
Degree $2$
Conductor $666$
Sign $-0.316 + 0.948i$
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 + (−0.448 − 1.67i)5-s + (0.633 − 1.09i)7-s + (0.707 − 0.707i)8-s + 1.73·10-s − 2.44·11-s + (0.366 − 0.0980i)13-s + (0.896 + 0.896i)14-s + (0.500 + 0.866i)16-s + (−6.24 − 1.67i)17-s + (−5.09 + 1.36i)19-s + (−0.448 + 1.67i)20-s + (0.633 − 2.36i)22-s + (−2.44 + 2.44i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.200 − 0.748i)5-s + (0.239 − 0.415i)7-s + (0.249 − 0.249i)8-s + 0.547·10-s − 0.738·11-s + (0.101 − 0.0272i)13-s + (0.239 + 0.239i)14-s + (0.125 + 0.216i)16-s + (−1.51 − 0.405i)17-s + (−1.16 + 0.313i)19-s + (−0.100 + 0.374i)20-s + (0.135 − 0.504i)22-s + (−0.510 + 0.510i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.316 + 0.948i$
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: \((2,\ 666,\ (\ :1/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318253 - 0.441462i\)
\(L(\frac12)\) \(\approx\) \(0.318253 - 0.441462i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.69 + 3.86i)T \)
good5 \( 1 + (0.448 + 1.67i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.633 + 1.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + (-0.366 + 0.0980i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (6.24 + 1.67i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.09 - 1.36i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.44 - 2.44i)T - 23iT^{2} \)
29 \( 1 + (6.12 + 6.12i)T + 29iT^{2} \)
31 \( 1 + (-5.73 + 5.73i)T - 31iT^{2} \)
41 \( 1 + (2.89 - 5.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.267 - 0.267i)T + 43iT^{2} \)
47 \( 1 + 4.24iT - 47T^{2} \)
53 \( 1 + (-5.22 + 3.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.79 - 1.55i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.86 + 10.6i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.63 + 2.09i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.12 + 1.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.39iT - 73T^{2} \)
79 \( 1 + (3.36 - 0.901i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.10 + 1.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.10 + 4.12i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.366 - 0.366i)T + 97iT^{2} \)
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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

−10.15811084900956790462747297145, −9.198049493956212662487223625003, −8.384689489570842420830098925828, −7.78095735257748501369530617884, −6.74674857767966815821958254550, −5.78690164049919864103163838394, −4.71618941877889700905297065031, −4.05111291404073753373981057168, −2.15202898439888007987714522803, −0.29157420529835669830939098201, 1.99381195249715373085267894109, 2.91476556795921281214558140393, 4.13494628923853959211672468714, 5.15693866801822024287442296485, 6.44902257247318876356956554427, 7.26537866432331529632725949643, 8.564630022989225841198221621650, 8.832852237725522020241453721344, 10.35354839015437447487699223493, 10.64400117104226275108570621856

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