Properties

Label 2-666-111.8-c1-0-5
Degree $2$
Conductor $666$
Sign $0.954 - 0.299i$
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 + (−0.205 − 0.765i)5-s + (−0.103 + 0.179i)7-s + (0.707 − 0.707i)8-s + 0.792·10-s − 2.15·11-s + (3.34 − 0.896i)13-s + (−0.146 − 0.146i)14-s + (0.500 + 0.866i)16-s + (1.47 + 0.394i)17-s + (5.53 − 1.48i)19-s + (−0.205 + 0.765i)20-s + (0.557 − 2.08i)22-s + (0.186 − 0.186i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.0917 − 0.342i)5-s + (−0.0392 + 0.0679i)7-s + (0.249 − 0.249i)8-s + 0.250·10-s − 0.649·11-s + (0.927 − 0.248i)13-s + (−0.0392 − 0.0392i)14-s + (0.125 + 0.216i)16-s + (0.357 + 0.0956i)17-s + (1.26 − 0.339i)19-s + (−0.0458 + 0.171i)20-s + (0.118 − 0.443i)22-s + (0.0388 − 0.0388i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.954 - 0.299i$
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: \((2,\ 666,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28856 + 0.197687i\)
\(L(\frac12)\) \(\approx\) \(1.28856 + 0.197687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.85 + 1.66i)T \)
good5 \( 1 + (0.205 + 0.765i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.103 - 0.179i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.47 - 0.394i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.53 + 1.48i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.186 + 0.186i)T - 23iT^{2} \)
29 \( 1 + (-0.667 - 0.667i)T + 29iT^{2} \)
31 \( 1 + (-2.39 + 2.39i)T - 31iT^{2} \)
41 \( 1 + (-2.63 + 4.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 + 1.86i)T + 43iT^{2} \)
47 \( 1 - 3.73iT - 47T^{2} \)
53 \( 1 + (-0.970 + 0.560i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.83 - 1.83i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.320 + 1.19i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (6.44 + 3.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.47 + 2.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.83iT - 73T^{2} \)
79 \( 1 + (3.34 - 0.896i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (7.29 - 4.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.507 + 1.89i)T + (-77.0 - 44.5i)T^{2} \)
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

−10.44489696379272165035665772484, −9.549033253182915427327253525079, −8.700638739911406072739135307307, −7.952683731861704378775896516433, −7.14347516677104479990505110890, −6.01046560914407803787256839334, −5.29112309808822471939653931098, −4.21487885426613697228918132046, −2.89140175964384526499300932136, −0.965167388416510890829821565988, 1.19653579886969583466019035596, 2.77016772600560596060052994269, 3.61979363978260849226077950824, 4.86283481457407018698415038595, 5.90261547659553930205679422450, 7.09536960753898469497521374674, 7.974537093362773741435163367188, 8.834778681150568085955833380234, 9.811720232628705696895979351638, 10.46106673037710504171826262259

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