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Properties

Label 2-66-11.9-c1-0-1
Degree $2$
Conductor $66$
Sign $0.675 + 0.737i$
Analytic cond. $0.527012$
Root an. cond. $0.725956$
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  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.881 + 2.71i)5-s + (−0.309 − 0.951i)6-s + (−3.73 − 2.71i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 2.85·10-s + (1.23 + 3.07i)11-s − 0.999·12-s + (−1 + 3.07i)13-s + (−3.73 + 2.71i)14-s + (2.30 + 1.67i)15-s + (0.309 + 0.951i)16-s + (−0.763 − 2.35i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.394 + 1.21i)5-s + (−0.126 − 0.388i)6-s + (−1.41 − 1.02i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + 0.902·10-s + (0.372 + 0.927i)11-s − 0.288·12-s + (−0.277 + 0.853i)13-s + (−0.998 + 0.725i)14-s + (0.596 + 0.433i)15-s + (0.0772 + 0.237i)16-s + (−0.185 − 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946344 - 0.416368i\)
\(L(\frac12)\) \(\approx\) \(0.946344 - 0.416368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-1.23 - 3.07i)T \)
good5 \( 1 + (-0.881 - 2.71i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.73 + 2.71i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1 - 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.763 + 2.35i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.61 + 1.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.23T + 23T^{2} \)
29 \( 1 + (-0.309 - 0.224i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.66 + 8.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.23 + 0.898i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.61 + 1.90i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + (2 - 1.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.19 - 3.66i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.11 - 2.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.09 + 6.43i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-3.85 - 11.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.16 - 6.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.88 - 8.86i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.208 + 0.640i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-3.5 + 10.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.24877257433065475041057434289, −13.73726511968201784921522542436, −12.65734239272869508996642640540, −11.35979307059915347956851765828, −9.972931815834157436577685010723, −9.556910968959328256918041132018, −7.23981222633069435622056849657, −6.50988425381615641140379066244, −3.98143876458090391076324401961, −2.58932707928477112464601696660, 3.33683025332866080474040301222, 5.25527585676081333804439890501, 6.24724824530057742156387244425, 8.285780652857303209955982758433, 9.041222480456437267817730434656, 9.997018121329256051312572490793, 12.20105422384352318508955667981, 12.93129864811880856625453294527, 13.87586881843360857721096294098, 15.22906561725887678253815752748

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