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Properties

Label 2-666-9.4-c1-0-19
Degree $2$
Conductor $666$
Sign $0.963 + 0.267i$
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 + (0.924 + 1.46i)3-s + (−0.499 + 0.866i)4-s + (0.606 − 1.04i)5-s + (0.806 − 1.53i)6-s + (−0.530 − 0.918i)7-s + 0.999·8-s + (−1.29 + 2.70i)9-s − 1.21·10-s + (0.424 + 0.734i)11-s + (−1.73 + 0.0679i)12-s + (2.49 − 4.32i)13-s + (−0.530 + 0.918i)14-s + (2.09 − 0.0823i)15-s + (−0.5 − 0.866i)16-s + 5.69·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.533 + 0.845i)3-s + (−0.249 + 0.433i)4-s + (0.271 − 0.469i)5-s + (0.329 − 0.625i)6-s + (−0.200 − 0.347i)7-s + 0.353·8-s + (−0.430 + 0.902i)9-s − 0.383·10-s + (0.127 + 0.221i)11-s + (−0.499 + 0.0196i)12-s + (0.692 − 1.19i)13-s + (−0.141 + 0.245i)14-s + (0.541 − 0.0212i)15-s + (−0.125 − 0.216i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.963 + 0.267i$
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: \((2,\ 666,\ (\ :1/2),\ 0.963 + 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56811 - 0.213484i\)
\(L(\frac12)\) \(\approx\) \(1.56811 - 0.213484i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.924 - 1.46i)T \)
37 \( 1 - T \)
good5 \( 1 + (-0.606 + 1.04i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.530 + 0.918i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.424 - 0.734i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.49 + 4.32i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.69T + 17T^{2} \)
19 \( 1 - 0.967T + 19T^{2} \)
23 \( 1 + (-1.50 + 2.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.94 - 6.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.39 + 2.41i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (4.39 - 7.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.83 + 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.55 - 9.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.06T + 53T^{2} \)
59 \( 1 + (-2.05 + 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.66 - 8.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.16 + 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.99T + 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 + (-2.17 - 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.24 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.94T + 89T^{2} \)
97 \( 1 + (1.78 + 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.28222015941851920757634597493, −9.824300634243681161201788629404, −8.813119409077523090691526262910, −8.260981553563062394605997667015, −7.25133766483203874946766395541, −5.66816307808635199512047941458, −4.83860132945692345373384925729, −3.61972809116535409493865111470, −2.91604590678205782234890023372, −1.19797815692082693954157708933, 1.26303441431372273770955099497, 2.64265430505328345899072208933, 3.85996893067252597369631752343, 5.50570954821534726631725525676, 6.38289883330687783257193160508, 6.94322357316370557029364181172, 7.963385559370213963770460315743, 8.678009211718821121525448659854, 9.481669686607575045425706899181, 10.28739774142892625774609713130

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