Properties

Label 2-77-11.3-c1-0-1
Degree $2$
Conductor $77$
Sign $0.739 - 0.673i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (−1.99 + 1.44i)2-s + (−0.5 − 1.53i)3-s + (1.26 − 3.88i)4-s + (2.80 + 2.03i)5-s + (3.22 + 2.34i)6-s + (−0.309 + 0.951i)7-s + (1.58 + 4.89i)8-s + (0.309 − 0.224i)9-s − 8.55·10-s + (2.91 + 1.57i)11-s − 6.60·12-s + (0.528 − 0.384i)13-s + (−0.762 − 2.34i)14-s + (1.73 − 5.33i)15-s + (−3.65 − 2.65i)16-s + (0.919 + 0.668i)17-s + ⋯
L(s)  = 1  + (−1.41 + 1.02i)2-s + (−0.288 − 0.888i)3-s + (0.631 − 1.94i)4-s + (1.25 + 0.911i)5-s + (1.31 + 0.957i)6-s + (−0.116 + 0.359i)7-s + (0.561 + 1.72i)8-s + (0.103 − 0.0748i)9-s − 2.70·10-s + (0.880 + 0.474i)11-s − 1.90·12-s + (0.146 − 0.106i)13-s + (−0.203 − 0.626i)14-s + (0.447 − 1.37i)15-s + (−0.913 − 0.663i)16-s + (0.223 + 0.162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524624 + 0.203115i\)
\(L(\frac12)\) \(\approx\) \(0.524624 + 0.203115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-2.91 - 1.57i)T \)
good2 \( 1 + (1.99 - 1.44i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.80 - 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-0.528 + 0.384i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.919 - 0.668i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.87 + 5.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + (1.41 - 4.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.26 - 1.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.135 - 0.418i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.82 + 5.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (0.186 + 0.575i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.94 - 5.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.523 + 1.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.54 + 4.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (-4.38 - 3.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.07 + 6.37i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.14 - 1.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.41 + 3.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.698T + 89T^{2} \)
97 \( 1 + (-12.0 + 8.73i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.78233938825352195553695616191, −13.86691786378767464010721161092, −12.46760864269831927070234115557, −10.88840283652130002826113649596, −9.814124713478736739395913060203, −8.999690556266380448842236767820, −7.39128558922212606943318200635, −6.58658744842989403185398835552, −5.91897077543004412185023096152, −1.83225717415857255392004335705, 1.66189493881005694540507173399, 4.01998150583811817044212011463, 5.91440232868672542697155407730, 8.050714965178102199036054866134, 9.293525355848351403247834798104, 9.810572521521794965715447585694, 10.62184632044375661839094100488, 11.82078075693188683894972515375, 12.94215819285903095537444412325, 14.19129499164814085241883577455

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