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Properties

Label 2-66-33.29-c1-0-0
Degree $2$
Conductor $66$
Sign $0.772 - 0.635i$
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  + (−0.809 − 0.587i)2-s + (−0.751 + 1.56i)3-s + (0.309 + 0.951i)4-s + (1.56 + 2.14i)5-s + (1.52 − 0.820i)6-s + (1.56 − 0.507i)7-s + (0.309 − 0.951i)8-s + (−1.86 − 2.34i)9-s − 2.65i·10-s + (−2.77 + 1.81i)11-s + (−1.71 − 0.232i)12-s + (0.885 − 1.21i)13-s + (−1.56 − 0.507i)14-s + (−4.52 + 0.820i)15-s + (−0.809 + 0.587i)16-s + (5.49 − 3.99i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.434 + 0.900i)3-s + (0.154 + 0.475i)4-s + (0.697 + 0.960i)5-s + (0.622 − 0.334i)6-s + (0.589 − 0.191i)7-s + (0.109 − 0.336i)8-s + (−0.623 − 0.782i)9-s − 0.839i·10-s + (−0.837 + 0.546i)11-s + (−0.495 − 0.0671i)12-s + (0.245 − 0.338i)13-s + (−0.417 − 0.135i)14-s + (−1.16 + 0.211i)15-s + (−0.202 + 0.146i)16-s + (1.33 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.772 - 0.635i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.772 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.648965 + 0.232583i\)
\(L(\frac12)\) \(\approx\) \(0.648965 + 0.232583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.751 - 1.56i)T \)
11 \( 1 + (2.77 - 1.81i)T \)
good5 \( 1 + (-1.56 - 2.14i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.56 + 0.507i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.885 + 1.21i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.49 + 3.99i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.21 + 1.69i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.78iT - 23T^{2} \)
29 \( 1 + (-0.143 - 0.441i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.20 - 3.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.311 - 0.957i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.486 - 1.49i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.42iT - 43T^{2} \)
47 \( 1 + (9.52 + 3.09i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.77 - 7.94i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.81 + 1.88i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.17 + 11.2i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + (0.527 + 0.726i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.40 - 1.42i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.86 - 2.57i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.51 + 1.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.8iT - 89T^{2} \)
97 \( 1 + (-2.90 - 2.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−15.00429619402193627374009904477, −14.12218880618046781479094613260, −12.52065636409206878970048416382, −11.16204225147927774367138250140, −10.42804416494141148442977144106, −9.721196043601058441613062118654, −8.133890211785773103627799536680, −6.52148379643624095244332515404, −4.88307782259944217957795320737, −2.86278413742470247620087148853, 1.62919667802991226540446627829, 5.27126503527407830053853240928, 6.12241772040636747407662715496, 7.88012044019202129765123307645, 8.574429501177036302357977205331, 10.13796602464428190461227398664, 11.39871236328205606335212137112, 12.66833266844377131423947824769, 13.51847315894141264915619263528, 14.70366157555010527012196976507

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