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Properties

Label 2-777-777.65-c0-0-0
Degree $2$
Conductor $777$
Sign $-0.143 - 0.989i$
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  + (−0.939 + 0.342i)3-s + (−0.766 − 0.642i)4-s − 7-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)12-s + (0.673 + 1.85i)13-s + (0.173 + 0.984i)16-s + (−1.11 + 1.32i)19-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 0.642i)28-s + (0.592 − 0.342i)31-s − 36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.766 − 0.642i)4-s − 7-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)12-s + (0.673 + 1.85i)13-s + (0.173 + 0.984i)16-s + (−1.11 + 1.32i)19-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 0.642i)28-s + (0.592 − 0.342i)31-s − 36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.143 - 0.989i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ -0.143 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3720296216\)
\(L(\frac12)\) \(\approx\) \(0.3720296216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + 0.684iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.57008465198323157020068653453, −9.915666829976862979858638965970, −9.305177857040408127375865314966, −8.469626671094307059361437379045, −6.87518531422060853260931088610, −6.24792613239786661224373416374, −5.56746701396616661728465704199, −4.28981678469545985136620235094, −3.83373276505690075670602368233, −1.60727652200769884972551269242, 0.45806896902558064508807683772, 2.76259999285446846418709928358, 3.88394317398880926729111227264, 4.94711107904113831967751849462, 5.89240315793775249092160838656, 6.69727813674763632698352667171, 7.73053919483997010492946099681, 8.479982768565164484013773894235, 9.501526482721713009162472049678, 10.37907576607918876387998157158

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