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Properties

Label 2-666-111.5-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.966 - 0.258i$
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  + (−0.906 − 0.422i)2-s + (0.642 + 0.766i)4-s + (−1.52 − 1.06i)5-s + (0.800 + 4.53i)7-s + (−0.258 − 0.965i)8-s + (0.928 + 1.60i)10-s + (0.382 − 0.662i)11-s + (−5.11 − 0.447i)13-s + (1.19 − 4.45i)14-s + (−0.173 + 0.984i)16-s + (1.69 − 0.148i)17-s + (−2.29 − 4.91i)19-s + (−0.161 − 1.85i)20-s + (−0.626 + 0.438i)22-s + (−3.89 − 1.04i)23-s + ⋯
L(s)  = 1  + (−0.640 − 0.298i)2-s + (0.321 + 0.383i)4-s + (−0.680 − 0.476i)5-s + (0.302 + 1.71i)7-s + (−0.0915 − 0.341i)8-s + (0.293 + 0.508i)10-s + (0.115 − 0.199i)11-s + (−1.41 − 0.124i)13-s + (0.318 − 1.18i)14-s + (−0.0434 + 0.246i)16-s + (0.411 − 0.0360i)17-s + (−0.526 − 1.12i)19-s + (−0.0362 − 0.413i)20-s + (−0.133 + 0.0935i)22-s + (−0.812 − 0.217i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.966 - 0.258i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.966 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0103679 + 0.0790033i\)
\(L(\frac12)\) \(\approx\) \(0.0103679 + 0.0790033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.906 + 0.422i)T \)
3 \( 1 \)
37 \( 1 + (1.35 - 5.92i)T \)
good5 \( 1 + (1.52 + 1.06i)T + (1.71 + 4.69i)T^{2} \)
7 \( 1 + (-0.800 - 4.53i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.382 + 0.662i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.11 + 0.447i)T + (12.8 + 2.25i)T^{2} \)
17 \( 1 + (-1.69 + 0.148i)T + (16.7 - 2.95i)T^{2} \)
19 \( 1 + (2.29 + 4.91i)T + (-12.2 + 14.5i)T^{2} \)
23 \( 1 + (3.89 + 1.04i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (9.12 - 2.44i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.11 + 1.11i)T - 31iT^{2} \)
41 \( 1 + (1.15 - 0.965i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.40 - 5.40i)T + 43iT^{2} \)
47 \( 1 + (4.91 - 2.83i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.79 + 1.55i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.19 + 5.99i)T + (-20.1 + 55.4i)T^{2} \)
61 \( 1 + (-0.804 + 9.19i)T + (-60.0 - 10.5i)T^{2} \)
67 \( 1 + (-6.71 + 1.18i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.79 - 7.69i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + 2.36iT - 73T^{2} \)
79 \( 1 + (3.40 - 4.86i)T + (-27.0 - 74.2i)T^{2} \)
83 \( 1 + (6.85 - 8.17i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.10 + 4.27i)T + (30.4 - 83.6i)T^{2} \)
97 \( 1 + (3.70 - 13.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.13286590245559347670948135413, −9.747090452633834805819245518252, −9.250510537412426474705372736315, −8.305399937176122180237372031093, −7.82949289224057476728421200008, −6.57332344723625174060100310325, −5.41485612222538306587925234734, −4.52928849818934049824393080921, −2.98885781124436398419281895399, −2.01399620444805025794217162497, 0.04959966814243772046920070577, 1.82018525207223802251229100898, 3.59871506324782623890827947245, 4.38369843948328803730098879604, 5.74169622595252122657561724796, 7.06512124161963495306923825459, 7.47393107358814806628364293566, 8.013702632956216675834543837580, 9.422791390747833739861584567042, 10.17578893358750061583927366013

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