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Properties

Label 2-66-33.17-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: $\chi_{66} (17, \cdot )$
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

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

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