Properties

Label 2-666-111.23-c1-0-5
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: $\chi_{666} (467, \cdot )$
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

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

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