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Properties

Label 2-666-9.7-c1-0-23
Degree $2$
Conductor $666$
Sign $0.630 + 0.776i$
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 + (1.57 − 0.719i)3-s + (−0.499 − 0.866i)4-s + (1.33 + 2.30i)5-s + (0.165 − 1.72i)6-s + (0.530 − 0.918i)7-s − 0.999·8-s + (1.96 − 2.26i)9-s + 2.66·10-s + (0.153 − 0.266i)11-s + (−1.41 − 1.00i)12-s + (2.73 + 4.74i)13-s + (−0.530 − 0.918i)14-s + (3.76 + 2.67i)15-s + (−0.5 + 0.866i)16-s + 2.15·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.909 − 0.415i)3-s + (−0.249 − 0.433i)4-s + (0.596 + 1.03i)5-s + (0.0673 − 0.703i)6-s + (0.200 − 0.347i)7-s − 0.353·8-s + (0.655 − 0.755i)9-s + 0.842·10-s + (0.0463 − 0.0803i)11-s + (−0.407 − 0.290i)12-s + (0.759 + 1.31i)13-s + (−0.141 − 0.245i)14-s + (0.970 + 0.691i)15-s + (−0.125 + 0.216i)16-s + 0.521·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.630 + 0.776i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.630 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42775 - 1.15636i\)
\(L(\frac12)\) \(\approx\) \(2.42775 - 1.15636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.719i)T \)
37 \( 1 + T \)
good5 \( 1 + (-1.33 - 2.30i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.530 + 0.918i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.153 + 0.266i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.73 - 4.74i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 + 1.58T + 19T^{2} \)
23 \( 1 + (4.41 + 7.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.54 - 2.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.159 + 0.276i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 + (1.16 + 2.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.224 + 0.388i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.50 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.65T + 53T^{2} \)
59 \( 1 + (3.35 + 5.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.40 + 5.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.27 + 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.59T + 71T^{2} \)
73 \( 1 - 6.20T + 73T^{2} \)
79 \( 1 + (4.06 - 7.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.41 - 2.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.49T + 89T^{2} \)
97 \( 1 + (-4.39 + 7.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.46435064896950224601137143428, −9.618342367601200185607803967283, −8.806968638135543586694470588389, −7.81157411025174521448813442684, −6.63329368230326962623454841790, −6.23521461465227226010813368793, −4.49310801962360955525952060923, −3.57688856030507493506173072982, −2.52900481241864692546031675406, −1.59235383544498290319512146419, 1.65395453216267639893483125193, 3.17496587556593395379304801574, 4.17549881200752666866147528763, 5.35406014164376677580988388018, 5.75947284490429681034339404083, 7.33462843756401603580137054229, 8.241460521366666199021850623364, 8.670329746345010591791250965224, 9.629248598266689788841139017634, 10.27563740844009865119331559886

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