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Properties

Label 2-666-333.85-c1-0-14
Degree $2$
Conductor $666$
Sign $-0.619 + 0.784i$
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  + (−0.866 − 0.5i)2-s + (−1.52 − 0.822i)3-s + (0.499 + 0.866i)4-s + (−2.98 + 1.72i)5-s + (0.908 + 1.47i)6-s − 0.480·7-s − 0.999i·8-s + (1.64 + 2.50i)9-s + 3.44·10-s + (1.57 + 2.73i)11-s + (−0.0497 − 1.73i)12-s + (1.16 − 0.675i)13-s + (0.415 + 0.240i)14-s + (5.96 − 0.171i)15-s + (−0.5 + 0.866i)16-s + (0.944 − 0.545i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.880 − 0.474i)3-s + (0.249 + 0.433i)4-s + (−1.33 + 0.770i)5-s + (0.371 + 0.601i)6-s − 0.181·7-s − 0.353i·8-s + (0.548 + 0.835i)9-s + 1.08·10-s + (0.476 + 0.824i)11-s + (−0.0143 − 0.499i)12-s + (0.324 − 0.187i)13-s + (0.111 + 0.0641i)14-s + (1.54 − 0.0442i)15-s + (−0.125 + 0.216i)16-s + (0.229 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.619 + 0.784i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.619 + 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113263 - 0.233704i\)
\(L(\frac12)\) \(\approx\) \(0.113263 - 0.233704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.52 + 0.822i)T \)
37 \( 1 + (-5.65 + 2.23i)T \)
good5 \( 1 + (2.98 - 1.72i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.480T + 7T^{2} \)
11 \( 1 + (-1.57 - 2.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 0.675i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.944 + 0.545i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.333 - 0.192i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.91 + 3.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.40 - 3.11i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.60 + 2.07i)T + (15.5 + 26.8i)T^{2} \)
41 \( 1 + (-1.65 - 2.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.47 + 4.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.93 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.47 - 2.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.340iT - 59T^{2} \)
61 \( 1 + 14.2iT - 61T^{2} \)
67 \( 1 + (-4.44 - 7.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.04 + 1.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.32T + 73T^{2} \)
79 \( 1 + 16.2iT - 79T^{2} \)
83 \( 1 + (-5.37 + 9.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.15 + 1.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 6.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.43674980823430858676556394808, −9.550113820424935662553490472274, −8.222616631526291508527119673725, −7.52439344560298209562286070545, −6.92216124300417595841380692199, −5.96721945911739869004864558147, −4.44221091998422942137358533022, −3.54301110019321600367962743780, −2.01160146646397819843976000139, −0.23158333970017059570859568856, 1.09986393833923784749156086959, 3.62568110381872531616114123827, 4.33171908676253464585091554053, 5.56685145720940005644575623140, 6.28290178098357794783233016013, 7.49388239647333167354164437707, 8.180895930836284180440025140320, 9.153001896418697646345564660523, 9.795361009541189808395917423187, 11.07879240812674914175629807302

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