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Properties

Label 2-666-111.29-c1-0-6
Degree $2$
Conductor $666$
Sign $0.999 - 0.0312i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (2.43 − 0.653i)5-s + (0.762 − 1.32i)7-s + (0.707 + 0.707i)8-s + 2.52·10-s + 0.293·11-s + (0.472 + 1.76i)13-s + (1.07 − 1.07i)14-s + (0.500 + 0.866i)16-s + (0.0537 − 0.200i)17-s + (−0.213 − 0.798i)19-s + (2.43 + 0.653i)20-s + (0.283 + 0.0760i)22-s + (−5.10 − 5.10i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (1.09 − 0.292i)5-s + (0.288 − 0.498i)7-s + (0.249 + 0.249i)8-s + 0.798·10-s + 0.0885·11-s + (0.130 + 0.488i)13-s + (0.288 − 0.288i)14-s + (0.125 + 0.216i)16-s + (0.0130 − 0.0486i)17-s + (−0.0490 − 0.183i)19-s + (0.545 + 0.146i)20-s + (0.0604 + 0.0162i)22-s + (−1.06 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.999 - 0.0312i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.999 - 0.0312i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73825 + 0.0427329i\)
\(L(\frac12)\) \(\approx\) \(2.73825 + 0.0427329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
37 \( 1 + (1.03 - 5.99i)T \)
good5 \( 1 + (-2.43 + 0.653i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.762 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.293T + 11T^{2} \)
13 \( 1 + (-0.472 - 1.76i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.0537 + 0.200i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.213 + 0.798i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.10 + 5.10i)T + 23iT^{2} \)
29 \( 1 + (-4.72 + 4.72i)T - 29iT^{2} \)
31 \( 1 + (-4.39 - 4.39i)T + 31iT^{2} \)
41 \( 1 + (0.398 - 0.691i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.60 - 5.60i)T - 43iT^{2} \)
47 \( 1 + 0.508iT - 47T^{2} \)
53 \( 1 + (3.09 - 1.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.715 + 2.66i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.820 + 0.219i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.106 + 0.0617i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.0 + 6.95i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.83iT - 73T^{2} \)
79 \( 1 + (0.472 + 1.76i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-4.50 + 2.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.2 + 2.74i)T + (77.0 + 44.5i)T^{2} \)
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

−10.41510488136709464027599622455, −9.849064628322217162407913367963, −8.721718328589700416304065855167, −7.87442573688262921528128216908, −6.61785275275997298304645749052, −6.12769952246120864087696381144, −4.94580466425950200026918856330, −4.23926480270077463339941042768, −2.76366478666720728245366436854, −1.53472770501670063893661504598, 1.69660280742526287324001808128, 2.68818514106345317193103903285, 3.91450192345939765090889080035, 5.25747029464136014382062452946, 5.81887091018177316899297599993, 6.67732406991567692871778655420, 7.84018379296176998077192832986, 8.879960270491189638222623390404, 9.928387549101410087007551611651, 10.41133019542699403306666023704

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