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Properties

Label 2-77-11.3-c1-0-2
Degree $2$
Conductor $77$
Sign $0.827 - 0.561i$
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  + (−0.183 + 0.132i)2-s + (−0.0677 − 0.208i)3-s + (−0.602 + 1.85i)4-s + (2.01 + 1.46i)5-s + (0.0401 + 0.0291i)6-s + (0.309 − 0.951i)7-s + (−0.276 − 0.849i)8-s + (2.38 − 1.73i)9-s − 0.564·10-s + (−2.66 − 1.97i)11-s + 0.427·12-s + (−4.15 + 3.01i)13-s + (0.0699 + 0.215i)14-s + (0.168 − 0.520i)15-s + (−2.98 − 2.17i)16-s + (1.16 + 0.844i)17-s + ⋯
L(s)  = 1  + (−0.129 + 0.0940i)2-s + (−0.0390 − 0.120i)3-s + (−0.301 + 0.926i)4-s + (0.902 + 0.655i)5-s + (0.0163 + 0.0118i)6-s + (0.116 − 0.359i)7-s + (−0.0975 − 0.300i)8-s + (0.796 − 0.578i)9-s − 0.178·10-s + (−0.803 − 0.594i)11-s + 0.123·12-s + (−1.15 + 0.837i)13-s + (0.0186 + 0.0574i)14-s + (0.0436 − 0.134i)15-s + (−0.747 − 0.543i)16-s + (0.282 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.827 - 0.561i$
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: \((2,\ 77,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877193 + 0.269311i\)
\(L(\frac12)\) \(\approx\) \(0.877193 + 0.269311i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.66 + 1.97i)T \)
good2 \( 1 + (0.183 - 0.132i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.0677 + 0.208i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.01 - 1.46i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.15 - 3.01i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.16 - 0.844i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.87 + 5.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.08T + 23T^{2} \)
29 \( 1 + (-2.01 + 6.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.22 - 4.51i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.23 - 3.78i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.08 - 6.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + (-2.08 - 6.42i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.32 - 3.86i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.888 + 2.73i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.691 - 0.502i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (3.65 + 2.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.58 - 14.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.98 - 1.44i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.81 + 1.32i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (-9.77 + 7.09i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.44610390307334660373307826033, −13.39413734981064543762843485587, −12.68115888775046602271150210017, −11.29433776680609926655651215806, −10.01490454930493047406810957727, −9.029495816131084636543544179286, −7.46260381673755740268921204618, −6.61566920394765218343031272766, −4.65188983933093131117510533483, −2.81582577645182690231715345674, 1.94291250384609553130849150420, 4.96976379617104682343345312135, 5.51263073655474710786976022598, 7.45515254926106799376963703506, 9.039225874188846865779564893978, 10.00795416557705318344259430255, 10.61593086686272353529369055208, 12.55616320082669686010250131394, 13.15757862798486371961372308112, 14.49012410760816248066551927222

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