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Properties

Label 2-666-9.4-c1-0-13
Degree $2$
Conductor $666$
Sign $0.986 + 0.163i$
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  + (−0.5 − 0.866i)2-s + (1.41 − 1.00i)3-s + (−0.499 + 0.866i)4-s + (−1.74 + 3.02i)5-s + (−1.57 − 0.723i)6-s + (−1.43 − 2.47i)7-s + 0.999·8-s + (0.995 − 2.82i)9-s + 3.49·10-s + (2.71 + 4.70i)11-s + (0.160 + 1.72i)12-s + (−1.81 + 3.14i)13-s + (−1.43 + 2.47i)14-s + (0.559 + 6.02i)15-s + (−0.5 − 0.866i)16-s + 4.84·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.816 − 0.577i)3-s + (−0.249 + 0.433i)4-s + (−0.780 + 1.35i)5-s + (−0.642 − 0.295i)6-s + (−0.540 − 0.936i)7-s + 0.353·8-s + (0.331 − 0.943i)9-s + 1.10·10-s + (0.819 + 1.41i)11-s + (0.0462 + 0.497i)12-s + (−0.503 + 0.871i)13-s + (−0.382 + 0.662i)14-s + (0.144 + 1.55i)15-s + (−0.125 − 0.216i)16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39713 - 0.114667i\)
\(L(\frac12)\) \(\approx\) \(1.39713 - 0.114667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.41 + 1.00i)T \)
37 \( 1 + T \)
good5 \( 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.43 + 2.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.71 - 4.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.81 - 3.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.84T + 17T^{2} \)
19 \( 1 - 7.31T + 19T^{2} \)
23 \( 1 + (-3.34 + 5.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.90 - 5.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.48 - 7.76i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (-0.330 + 0.571i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.88 - 4.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.70 - 6.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.88T + 53T^{2} \)
59 \( 1 + (0.782 - 1.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.24 + 7.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.185 - 0.321i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.584T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.42 + 2.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.31T + 89T^{2} \)
97 \( 1 + (6.41 + 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31058625985739434785911515318, −9.771517131721641902239574279034, −8.924394605085166162959759845176, −7.51130822825901969318669177747, −7.22206923908932397745308149480, −6.71280922001146503796294053942, −4.46872965591701100703466060216, −3.49096901445870665068090421653, −2.87625005222721337765939434940, −1.34883856785028758844248695682, 0.932013565805815905743477714896, 3.07792127890063403872553009489, 3.92579193300912308887585462947, 5.39786468528094352262762924403, 5.61359858826853520372939162946, 7.53289193637281685031272502078, 7.983535597643077156388644626594, 9.006763274266662422581981555070, 9.189936829334496374921773100558, 10.10512761828052494880903234910

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