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Properties

Label 2-777-777.380-c0-0-1
Degree $2$
Conductor $777$
Sign $-0.308 + 0.951i$
Analytic cond. $0.387773$
Root an. cond. $0.622714$
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  + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s − i·5-s + (0.5 − 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.499i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s − i·17-s − 19-s + (0.866 + 0.5i)20-s + (−0.866 + 0.499i)21-s + (0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s − i·5-s + (0.5 − 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.499i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s − i·17-s − 19-s + (0.866 + 0.5i)20-s + (−0.866 + 0.499i)21-s + (0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.308 + 0.951i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ -0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5212103460\)
\(L(\frac12)\) \(\approx\) \(0.5212103460\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + iT - T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.32350060443496011614396793480, −9.422653236002718638508526340619, −8.276350762472382693616548213267, −7.65916190360366941571963535474, −7.10745389099363132507787314084, −5.54300565071603911879697092825, −4.83444524397548115425220583358, −4.19627026295205843633634145217, −2.43812497263658536847180219670, −0.61152011865866723172846053284, 1.90406015610176458783964682047, 3.43406445621850261466993595025, 4.84708022364226399795578856375, 5.27528959259234390773992176214, 6.35959014456049004219378487943, 6.89666473506290891195693442860, 8.545894871839929071345839173816, 9.074533275919605798054004625665, 10.35759526148298239601359949387, 10.60850091979478773460287061062

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