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Properties

Label 2-666-111.14-c1-0-2
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: $\chi_{666} (125, \cdot )$
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

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

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