Properties

Label 2-77-77.68-c1-0-2
Degree $2$
Conductor $77$
Sign $0.410 - 0.911i$
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  + (−0.0202 + 0.0181i)2-s + (−0.500 + 1.12i)3-s + (−0.208 + 1.98i)4-s + (−0.240 + 1.13i)5-s + (−0.0103 − 0.0318i)6-s + (−0.296 − 2.62i)7-s + (−0.0639 − 0.0879i)8-s + (0.993 + 1.10i)9-s + (−0.0157 − 0.0272i)10-s + (3.28 − 0.453i)11-s + (−2.13 − 1.23i)12-s + (1.77 − 5.45i)13-s + (0.0538 + 0.0477i)14-s + (−1.15 − 0.837i)15-s + (−3.90 − 0.830i)16-s + (−4.17 + 4.64i)17-s + ⋯
L(s)  = 1  + (−0.0142 + 0.0128i)2-s + (−0.289 + 0.649i)3-s + (−0.104 + 0.994i)4-s + (−0.107 + 0.506i)5-s + (−0.00422 − 0.0129i)6-s + (−0.112 − 0.993i)7-s + (−0.0225 − 0.0311i)8-s + (0.331 + 0.367i)9-s + (−0.00497 − 0.00861i)10-s + (0.990 − 0.136i)11-s + (−0.615 − 0.355i)12-s + (0.491 − 1.51i)13-s + (0.0143 + 0.0127i)14-s + (−0.297 − 0.216i)15-s + (−0.977 − 0.207i)16-s + (−1.01 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.410 - 0.911i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723352 + 0.467735i\)
\(L(\frac12)\) \(\approx\) \(0.723352 + 0.467735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.296 + 2.62i)T \)
11 \( 1 + (-3.28 + 0.453i)T \)
good2 \( 1 + (0.0202 - 0.0181i)T + (0.209 - 1.98i)T^{2} \)
3 \( 1 + (0.500 - 1.12i)T + (-2.00 - 2.22i)T^{2} \)
5 \( 1 + (0.240 - 1.13i)T + (-4.56 - 2.03i)T^{2} \)
13 \( 1 + (-1.77 + 5.45i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.17 - 4.64i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (0.247 + 2.35i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-2.75 + 4.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.64 - 3.63i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.414 + 1.94i)T + (-28.3 + 12.6i)T^{2} \)
37 \( 1 + (-2.53 + 1.12i)T + (24.7 - 27.4i)T^{2} \)
41 \( 1 + (1.14 - 0.828i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.93iT - 43T^{2} \)
47 \( 1 + (-4.63 + 0.487i)T + (45.9 - 9.77i)T^{2} \)
53 \( 1 + (8.74 - 1.85i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (8.79 + 0.924i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 + (1.94 + 0.414i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (5.00 + 8.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.143 - 0.440i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0315 + 0.300i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (4.78 - 4.30i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (1.25 + 3.87i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-12.9 - 7.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.66 - 1.51i)T + (78.4 + 57.0i)T^{2} \)
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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

−14.82895349996725051388385674301, −13.39336436738572746881455651612, −12.71567239050838310748269940068, −10.99012587136207388885033110359, −10.65225311569428737664487669541, −9.021457023527535552726313768871, −7.73324777133592166092384527087, −6.58151321453978462301478006573, −4.50523063257054809696697035521, −3.42896824305288365483572462768, 1.61435932612981649893571003354, 4.52641294715188761318960487773, 6.05588595703891220732782205762, 6.90026395426232608222931168710, 9.033637398431617607399846108015, 9.412625420631831193523668163875, 11.36780703084509800286987060436, 11.95619556708097370827291254097, 13.21713701048569958478460335348, 14.25092294879009047998342587769

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