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Properties

Label 2-77-77.41-c1-0-2
Degree $2$
Conductor $77$
Sign $0.965 + 0.260i$
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  + (0.0784 + 0.107i)2-s + (0.612 − 1.88i)4-s + (2.51 + 0.817i)7-s + (0.505 − 0.164i)8-s + (−2.42 + 1.76i)9-s + (−3.17 + 0.964i)11-s + (0.109 + 0.335i)14-s + (−3.14 − 2.28i)16-s + (−0.380 − 0.123i)18-s + (−0.353 − 0.266i)22-s − 7.50·23-s + (1.54 + 4.75i)25-s + (3.08 − 4.24i)28-s + (9.26 + 3.00i)29-s − 1.58i·32-s + ⋯
L(s)  = 1  + (0.0554 + 0.0763i)2-s + (0.306 − 0.942i)4-s + (0.951 + 0.309i)7-s + (0.178 − 0.0580i)8-s + (−0.809 + 0.587i)9-s + (−0.956 + 0.290i)11-s + (0.0291 + 0.0897i)14-s + (−0.787 − 0.572i)16-s + (−0.0897 − 0.0291i)18-s + (−0.0752 − 0.0569i)22-s − 1.56·23-s + (0.309 + 0.951i)25-s + (0.582 − 0.801i)28-s + (1.71 + 0.558i)29-s − 0.279i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.965 + 0.260i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00348 - 0.133026i\)
\(L(\frac12)\) \(\approx\) \(1.00348 - 0.133026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.51 - 0.817i)T \)
11 \( 1 + (3.17 - 0.964i)T \)
good2 \( 1 + (-0.0784 - 0.107i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.50T + 23T^{2} \)
29 \( 1 + (-9.26 - 3.00i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.53 + 7.80i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.51 + 1.09i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 + (7.95 + 5.78i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.78 - 6.58i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 + 92.2i)T^{2} \)
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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

−14.42117578051802201394729031192, −13.71303743735996207380342918012, −12.10339013089304671288802135447, −11.01646584939254263259964452458, −10.25268379746485251908701530406, −8.692829354909916292704601794825, −7.51441586210811460258795905405, −5.80939066583853134185975395579, −4.91626339372464437211532315016, −2.23595303929477303313458774310, 2.78303236013012806077080009213, 4.48764994586875992624151151943, 6.26079259088659468758450560814, 7.897764726437214478279507423520, 8.446782370757037695061572092537, 10.28268257836641928879066018447, 11.47848009559158275932807556210, 12.16246448783503381143438833585, 13.48892674878230488961995011531, 14.40490204078801245342558493044

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