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Properties

Label 2-77-77.54-c1-0-0
Degree $2$
Conductor $77$
Sign $0.797 - 0.603i$
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  + (−1.43 − 0.830i)2-s + (−1.97 + 1.13i)3-s + (0.380 + 0.658i)4-s + (2.80 + 1.61i)5-s + 3.78·6-s + (2.23 + 1.41i)7-s + 2.05i·8-s + (1.09 − 1.88i)9-s + (−2.68 − 4.65i)10-s + (−3.17 + 0.955i)11-s + (−1.50 − 0.866i)12-s + 0.904·13-s + (−2.04 − 3.89i)14-s − 7.36·15-s + (2.47 − 4.28i)16-s + (1.78 + 3.08i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.587i)2-s + (−1.13 + 0.657i)3-s + (0.190 + 0.329i)4-s + (1.25 + 0.723i)5-s + 1.54·6-s + (0.844 + 0.534i)7-s + 0.727i·8-s + (0.363 − 0.629i)9-s + (−0.849 − 1.47i)10-s + (−0.957 + 0.287i)11-s + (−0.433 − 0.249i)12-s + 0.250·13-s + (−0.545 − 1.04i)14-s − 1.90·15-s + (0.617 − 1.07i)16-s + (0.432 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.797 - 0.603i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.797 - 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467285 + 0.156953i\)
\(L(\frac12)\) \(\approx\) \(0.467285 + 0.156953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.23 - 1.41i)T \)
11 \( 1 + (3.17 - 0.955i)T \)
good2 \( 1 + (1.43 + 0.830i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.97 - 1.13i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.904T + 13T^{2} \)
17 \( 1 + (-1.78 - 3.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.99 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.09 - 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 + (-0.358 + 0.207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 2.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 + 2.15iT - 43T^{2} \)
47 \( 1 + (3.11 + 1.79i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.39 + 7.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.29 + 5.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.97 - 3.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.660 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + (3.67 + 6.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.67 + 3.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + (2.41 + 1.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.82iT - 97T^{2} \)
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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

−14.69979663924930987970462915407, −13.51899066182403063716652460610, −11.76781316237475845786410035003, −10.99299557283573449219672063435, −10.22508481337559119901202478273, −9.532339835444185491952927556900, −7.971858066674530158674138626983, −5.92083037430603253126738443195, −5.17539859208302699595968189149, −2.17603590953357954119323265445, 1.15689358901147037513307872455, 5.11617450722303836046761884750, 6.14833978327262641692158142117, 7.44558224735066591115251271400, 8.529584583284455712555804283023, 9.868217784288184485220089704868, 10.82273807290923731320150670283, 12.29537582172047931482106675690, 13.16415159473546565309366592856, 14.24779681846190993602198383355

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