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Properties

Label 2-2664-2664.1627-c0-0-8
Degree $2$
Conductor $2664$
Sign $-0.997 - 0.0697i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
Motivic weight $0$
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 + (0.309 + 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (0.978 + 0.207i)6-s − 0.999·8-s + (−0.809 + 0.587i)9-s − 1.61·10-s + (0.104 − 0.181i)11-s + (0.669 − 0.743i)12-s + (−0.978 − 1.69i)13-s + (1.08 − 1.20i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)18-s + (−0.809 + 1.40i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.309 + 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (0.978 + 0.207i)6-s − 0.999·8-s + (−0.809 + 0.587i)9-s − 1.61·10-s + (0.104 − 0.181i)11-s + (0.669 − 0.743i)12-s + (−0.978 − 1.69i)13-s + (1.08 − 1.20i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)18-s + (−0.809 + 1.40i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.997 - 0.0697i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ -0.997 - 0.0697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7377477879\)
\(L(\frac12)\) \(\approx\) \(0.7377477879\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + T \)
good5 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.104 + 0.181i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.978 + 1.69i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.669 - 1.15i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.978 - 1.69i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.913 + 1.58i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.209T + T^{2} \)
79 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.973867243513517661232743811460, −8.116431325730820265547297800561, −7.42735658366170388236890811491, −5.69123749485070751656888932915, −5.14567224547635529360986498998, −4.79851247708633225951509546469, −3.54222145757376375999772435805, −3.39886045127905311738490455685, −1.90512145991574026194646274759, −0.36592648421615334169519745721, 2.17410898921313512216720859954, 2.96277415362287566625268236106, 3.86242863986529287022263680697, 4.65421266311294312497095285270, 5.90553512470438406699376963814, 6.72982655807128968237240318451, 7.07841089127807594081823678237, 7.48082333061912539771300756200, 8.429615106178552535114094036492, 9.072361568064547331964403747495

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