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Properties

Label 2-66-33.17-c1-0-1
Degree $2$
Conductor $66$
Sign $0.990 + 0.140i$
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 + 1.53i)3-s + (−0.809 + 0.587i)4-s + (2.01 + 0.654i)5-s + (1.20 − 1.24i)6-s + (−2.01 − 2.77i)7-s + (0.809 + 0.587i)8-s + (−1.69 + 2.47i)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 + (0.627 + 3.61i)15-s + (0.309 − 0.951i)16-s + (−1.59 + 4.90i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.467 + 0.884i)3-s + (−0.404 + 0.293i)4-s + (0.901 + 0.292i)5-s + (0.492 − 0.507i)6-s + (−0.761 − 1.04i)7-s + (0.286 + 0.207i)8-s + (−0.563 + 0.826i)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 + (0.162 + 0.933i)15-s + (0.0772 − 0.237i)16-s + (−0.386 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.990 + 0.140i$
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: \((2,\ 66,\ (\ :1/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.911742 - 0.0641903i\)
\(L(\frac12)\) \(\approx\) \(0.911742 - 0.0641903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 1.53i)T \)
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

−14.53362255604027215043450613543, −13.76074950210589180364167550305, −12.85178316070218042861346810112, −10.93913932258914663951930613384, −10.31023073761573116493001766004, −9.467961121786934721583720489219, −8.159490866525574171169424969804, −6.21735922517882028244322826516, −4.28287485542534995594607554011, −2.81831191006726850516452605326, 2.42739768799940196490023818891, 5.35047617730000454995995473419, 6.50768885062979735911828531145, 7.69843519765041325587362627678, 9.211544853327416493715520534437, 9.659929597977998214072477151180, 11.94676360555182959220623544619, 12.96350177487858256821769554257, 13.70581714276151717007855766892, 14.94539326093677547925424745398

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