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Properties

Label 2-666-9.4-c1-0-29
Degree $2$
Conductor $666$
Sign $0.939 + 0.342i$
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 + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (2 − 3.46i)5-s + (1.5 + 0.866i)6-s − 0.999·8-s + (1.5 − 2.59i)9-s + 3.99·10-s + (0.5 + 0.866i)11-s + 1.73i·12-s + (−3 + 5.19i)13-s − 6.92i·15-s + (−0.5 − 0.866i)16-s + 3·17-s + 3·18-s − 19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.894 − 1.54i)5-s + (0.612 + 0.353i)6-s − 0.353·8-s + (0.5 − 0.866i)9-s + 1.26·10-s + (0.150 + 0.261i)11-s + 0.499i·12-s + (−0.832 + 1.44i)13-s − 1.78i·15-s + (−0.125 − 0.216i)16-s + 0.727·17-s + 0.707·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.939 + 0.342i$
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: \((2,\ 666,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59343 - 0.457291i\)
\(L(\frac12)\) \(\approx\) \(2.59343 - 0.457291i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
37 \( 1 - T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (1.5 + 2.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.943595821880462257716174727871, −9.337688272681026416270595396808, −8.743354150476671103794894259464, −7.933934640929962433674461122829, −6.89855801716974714524573411143, −6.07381987054016463485060705327, −4.85724181469975152333303464075, −4.24925680893428164946973447359, −2.54565964853089336066968235778, −1.35402803091721531811731018178, 1.97478809787743220548822708940, 3.04116001306905430402924089523, 3.42131205082774140573758334602, 5.05572049751112669092680726517, 5.87841103485098055376736677230, 7.12217485441404289734932781738, 7.908991198085706424095052511726, 9.252241059919096688362242825717, 9.873016975895661991076048275966, 10.55082951457741611957993130786

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