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Properties

Label 2-666-333.232-c1-0-11
Degree $2$
Conductor $666$
Sign $-0.995 - 0.0946i$
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 + (1.08 + 1.34i)3-s + (−0.499 + 0.866i)4-s + (0.563 − 0.976i)5-s + (−0.622 + 1.61i)6-s − 4.63·7-s − 0.999·8-s + (−0.630 + 2.93i)9-s + 1.12·10-s + (−0.811 + 1.40i)11-s + (−1.71 + 0.268i)12-s + (−2.49 + 4.32i)13-s + (−2.31 − 4.01i)14-s + (1.92 − 0.303i)15-s + (−0.5 − 0.866i)16-s + (−0.427 + 0.740i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.628 + 0.777i)3-s + (−0.249 + 0.433i)4-s + (0.252 − 0.436i)5-s + (−0.254 + 0.659i)6-s − 1.75·7-s − 0.353·8-s + (−0.210 + 0.977i)9-s + 0.356·10-s + (−0.244 + 0.423i)11-s + (−0.493 + 0.0776i)12-s + (−0.691 + 1.19i)13-s + (−0.619 − 1.07i)14-s + (0.498 − 0.0782i)15-s + (−0.125 − 0.216i)16-s + (−0.103 + 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.995 - 0.0946i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.995 - 0.0946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0624069 + 1.31592i\)
\(L(\frac12)\) \(\approx\) \(0.0624069 + 1.31592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 - 1.34i)T \)
37 \( 1 + (-3.76 - 4.77i)T \)
good5 \( 1 + (-0.563 + 0.976i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 + (0.811 - 1.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.49 - 4.32i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.427 - 0.740i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.44 + 5.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.77 - 6.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.738 + 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.94 - 6.82i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 + (1.59 - 2.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.01 + 10.4i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.61 + 2.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.44 + 9.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 6.50T + 61T^{2} \)
67 \( 1 + (-2.32 + 4.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0533 - 0.0924i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 + 5.98T + 79T^{2} \)
83 \( 1 + (-3.53 - 6.13i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.97 - 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.23 + 5.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70876899030054815037778114257, −9.627048510020486402398484005323, −9.329822809440574477437726363032, −8.577039242713152751221139397520, −7.16092603081293901710836016500, −6.67072567601969902932824138612, −5.33674540600701125392351639689, −4.55251485664029215194552483858, −3.50069254003427112668155099208, −2.49718133861076231142368921641, 0.54775965698604951666192657059, 2.66720750232814214706815387868, 2.86764257171028623149354128629, 4.13725473806122659923349539278, 5.94107448266507921031908666136, 6.30156705363852686606018398246, 7.41124019058084047786895538390, 8.425980346147160316518505731226, 9.415083440450456462603400785675, 10.14064842770921404821300090928

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