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Properties

Label 2-77-77.69-c0-0-0
Degree $2$
Conductor $77$
Sign $0.822 - 0.568i$
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 + 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.190 + 0.587i)14-s + (0.190 − 0.587i)18-s + 0.618·22-s + 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (−0.5 + 1.53i)29-s + 32-s + (−0.190 − 0.587i)36-s + (0.190 − 0.587i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.190 + 0.587i)14-s + (0.190 − 0.587i)18-s + 0.618·22-s + 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (−0.5 + 1.53i)29-s + 32-s + (−0.190 − 0.587i)36-s + (0.190 − 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(0.0384280\)
Root analytic conductor: \(0.196030\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :0),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3940077207\)
\(L(\frac12)\) \(\approx\) \(0.3940077207\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−14.88908258775139866351665192359, −13.65532817676417750890003425419, −12.95006863085824755145486164698, −11.38933812356431967330611248854, −10.45975473868778616407928874431, −8.935139995367423919312740922857, −7.997295040882863506221465232122, −7.03454853540565329256094827597, −5.16992084082465797227471799327, −3.35405452155751025316205560791, 2.48329704472032446623488429724, 5.02305203641943809312723389715, 6.14991031457579889100996880776, 8.145878264529248582217719935070, 9.096411228983060337014424351550, 10.12551310524947616653456690813, 11.33219012771395860620261866981, 12.19464558618417044293041407130, 13.64000830322253703856461079702, 14.91028726419622096218462723070

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