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Properties

Label 2-77-77.20-c0-0-0
Degree $2$
Conductor $77$
Sign $-0.0457 - 0.998i$
Analytic cond. $0.0384280$
Root an. cond. $0.196030$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 1.53i)2-s + (−1.30 − 0.951i)4-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (1.30 − 0.951i)14-s + (1.30 + 0.951i)18-s − 1.61·22-s − 1.61·23-s + (−0.809 + 0.587i)25-s + (0.5 + 1.53i)28-s + (−0.5 − 0.363i)29-s + 0.999·32-s + (−1.30 + 0.951i)36-s + (1.30 + 0.951i)37-s + ⋯
L(s)  = 1  + (−0.5 + 1.53i)2-s + (−1.30 − 0.951i)4-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (1.30 − 0.951i)14-s + (1.30 + 0.951i)18-s − 1.61·22-s − 1.61·23-s + (−0.809 + 0.587i)25-s + (0.5 + 1.53i)28-s + (−0.5 − 0.363i)29-s + 0.999·32-s + (−1.30 + 0.951i)36-s + (1.30 + 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.0457 - 0.998i$
Analytic conductor: \(0.0384280\)
Root analytic conductor: \(0.196030\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :0),\ -0.0457 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3902853021\)
\(L(\frac12)\) \(\approx\) \(0.3902853021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−15.26177077717334900807571989157, −14.35074948394231895232406990142, −13.17855337388067672444338673541, −11.92155800903954175540221450599, −9.911879241447849028762294713107, −9.414322799362812733857412722582, −7.86864847338819353586449231672, −6.88058601121399683445656354047, −5.99237450344020077548331918694, −4.11400874670323536123765252069, 2.35278045266079068339323690453, 3.89149719521873110298670825170, 5.98104918423109067487880098106, 8.036248377990937902874141406961, 9.212076943517285374974722917022, 10.14236648571518692718561215438, 11.14297921968316790348669077014, 12.12127325306404572877805257579, 13.05999765204023432276871429946, 14.00574811998758571663709470627

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