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Properties

Label 2-66-33.2-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: $\chi_{66} (35, \cdot )$
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

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

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