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Properties

Label 2-666-37.3-c1-0-13
Degree $2$
Conductor $666$
Sign $-0.507 + 0.861i$
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  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (1.39 − 3.83i)5-s + (1.12 + 0.409i)7-s + (−0.866 − 0.500i)8-s + (−2.04 − 3.53i)10-s + (1.43 − 2.47i)11-s + (−2.53 + 0.446i)13-s + (1.03 − 0.599i)14-s + (−0.939 + 0.342i)16-s + (−2.55 − 0.450i)17-s + (5.32 + 6.34i)19-s + (−4.02 − 0.708i)20-s + (−0.978 − 2.68i)22-s + (−0.805 + 0.465i)23-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.0868 − 0.492i)4-s + (0.624 − 1.71i)5-s + (0.425 + 0.154i)7-s + (−0.306 − 0.176i)8-s + (−0.645 − 1.11i)10-s + (0.431 − 0.747i)11-s + (−0.702 + 0.123i)13-s + (0.277 − 0.160i)14-s + (−0.234 + 0.0855i)16-s + (−0.619 − 0.109i)17-s + (1.22 + 1.45i)19-s + (−0.898 − 0.158i)20-s + (−0.208 − 0.573i)22-s + (−0.168 + 0.0970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02962 - 1.80067i\)
\(L(\frac12)\) \(\approx\) \(1.02962 - 1.80067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 \)
37 \( 1 + (-4.74 - 3.80i)T \)
good5 \( 1 + (-1.39 + 3.83i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.12 - 0.409i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.43 + 2.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 - 0.446i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.55 + 0.450i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-5.32 - 6.34i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (0.805 - 0.465i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.633 + 0.365i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.74iT - 31T^{2} \)
41 \( 1 + (-2.19 - 12.4i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (2.96 + 5.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.16 - 1.51i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.03 - 2.83i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-12.1 + 2.13i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.52 - 0.556i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.98 + 3.34i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 + (3.28 - 9.01i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.56 + 8.89i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-1.64 - 4.50i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.05 + 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.991828473946498372005779736853, −9.536424534181695827930701567429, −8.624964438620578452938396277120, −7.88566084749434754855726894106, −6.27277183003716984322466951788, −5.41364086719015611386361732950, −4.79465035404542552637802875673, −3.72052242635854131799309483428, −2.09039269761788992759043628455, −1.02667948398906792988854899125, 2.21600376046668837423280820348, 3.12854817843232506330054064806, 4.43663807681944945541336563003, 5.46141964663456279827342162077, 6.57924485957290782422599568212, 7.07689663233003266745424631100, 7.76948048294711787594684817103, 9.268897821382541238953241400674, 9.863291098127139354133195367450, 10.98693999527328225236327579144

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