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Properties

Label 2-66-33.29-c1-0-1
Degree $2$
Conductor $66$
Sign $0.575 - 0.817i$
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 + (−0.309 + 1.70i)3-s + (0.309 + 0.951i)4-s + (−1.56 − 2.14i)5-s + (−1.25 + 1.19i)6-s + (1.56 − 0.507i)7-s + (−0.309 + 0.951i)8-s + (−2.80 − 1.05i)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 + (4.14 − 1.99i)15-s + (−0.809 + 0.587i)16-s + (−5.49 + 3.99i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.178 + 0.983i)3-s + (0.154 + 0.475i)4-s + (−0.697 − 0.960i)5-s + (−0.511 + 0.488i)6-s + (0.589 − 0.191i)7-s + (−0.109 + 0.336i)8-s + (−0.936 − 0.351i)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 + (1.06 − 0.515i)15-s + (−0.202 + 0.146i)16-s + (−1.33 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.575 - 0.817i$
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: \((2,\ 66,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939339 + 0.487738i\)
\(L(\frac12)\) \(\approx\) \(0.939339 + 0.487738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 1.70i)T \)
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

−15.28109638352037169803250996462, −14.13946979644454869657380637774, −12.86744872520684098390063087528, −11.65342421331855202197134384163, −10.82694344491583664076336575398, −8.946447571907939153667553667306, −8.244063326957215530040120796565, −6.24999259965489034782801917360, −4.74501425401033814928277335406, −3.91521803089721800676724414689, 2.35287665454615504372213505347, 4.35525222567087590118200143913, 6.35211795798774178720427325147, 7.21721794950996030856036161896, 8.747116906695869674539095999950, 10.75281274662684178244952843805, 11.53084839913653444807881699791, 12.30674635580421142849553924702, 13.62092573637823103099487596466, 14.54504737161995887503686781945

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