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Properties

Label 2-66-33.2-c1-0-2
Degree $2$
Conductor $66$
Sign $0.716 + 0.697i$
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 + (1.70 − 0.296i)3-s + (−0.809 − 0.587i)4-s + (−2.01 + 0.654i)5-s + (0.245 − 1.71i)6-s + (−2.01 + 2.77i)7-s + (−0.809 + 0.587i)8-s + (2.82 − 1.01i)9-s + 2.11i·10-s + (0.960 − 3.17i)11-s + (−1.55 − 0.763i)12-s + (−1.79 − 0.583i)13-s + (2.01 + 2.77i)14-s + (−3.24 + 1.71i)15-s + (0.309 + 0.951i)16-s + (1.59 + 4.90i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.985 − 0.170i)3-s + (−0.404 − 0.293i)4-s + (−0.901 + 0.292i)5-s + (0.100 − 0.699i)6-s + (−0.761 + 1.04i)7-s + (−0.286 + 0.207i)8-s + (0.941 − 0.336i)9-s + 0.670i·10-s + (0.289 − 0.957i)11-s + (−0.448 − 0.220i)12-s + (−0.497 − 0.161i)13-s + (0.538 + 0.741i)14-s + (−0.838 + 0.442i)15-s + (0.0772 + 0.237i)16-s + (0.386 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993549 - 0.403993i\)
\(L(\frac12)\) \(\approx\) \(0.993549 - 0.403993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.296i)T \)
11 \( 1 + (-0.960 + 3.17i)T \)
good5 \( 1 + (2.01 - 0.654i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.01 - 2.77i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.79 + 0.583i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.59 - 4.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.141 + 0.194i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.97iT - 23T^{2} \)
29 \( 1 + (3.37 + 2.45i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.02 + 6.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.14 - 5.18i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.42 - 4.66i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.51iT - 43T^{2} \)
47 \( 1 + (-1.98 - 2.72i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.91 - 1.92i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.62 - 2.24i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.53 + 0.500i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.98T + 67T^{2} \)
71 \( 1 + (9.47 - 3.07i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.45 - 2.00i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-10.7 - 3.50i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.12 - 6.53i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.47iT - 89T^{2} \)
97 \( 1 + (4.08 - 12.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.87938581446179670917696081498, −13.47712428732176398358837910124, −12.52890790928234256708474118514, −11.61938888837628164930349728661, −10.13360651100202288965410629743, −8.938750292907345518863854397397, −7.956375191856516954003681456785, −6.16679539252123391647183713585, −3.91629735374605768206944882359, −2.74555381041089698962566377988, 3.53151195006935578886229530422, 4.64619112168411022667491533389, 7.11749824762674865421816730714, 7.60019048249734644059207591644, 9.151770663618778225897351258195, 10.07673207860356387415994604366, 11.98727701620178368051006618724, 13.09853414736060976082731555151, 14.04282282994043367221455091142, 15.01955146845528188079873717843

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