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Properties

Label 2-777-777.221-c0-0-5
Degree $2$
Conductor $777$
Sign $0.444 + 0.895i$
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  + (−0.258 − 0.448i)2-s + (0.5 − 0.866i)3-s + (0.366 − 0.633i)4-s + (0.707 + 1.22i)5-s − 0.517·6-s + i·7-s − 0.896·8-s + (−0.499 − 0.866i)9-s + (0.366 − 0.633i)10-s + (−0.366 − 0.633i)12-s + (0.448 − 0.258i)14-s + 1.41·15-s + (−0.133 − 0.232i)16-s + (0.965 − 1.67i)17-s + (−0.258 + 0.448i)18-s + ⋯
L(s)  = 1  + (−0.258 − 0.448i)2-s + (0.5 − 0.866i)3-s + (0.366 − 0.633i)4-s + (0.707 + 1.22i)5-s − 0.517·6-s + i·7-s − 0.896·8-s + (−0.499 − 0.866i)9-s + (0.366 − 0.633i)10-s + (−0.366 − 0.633i)12-s + (0.448 − 0.258i)14-s + 1.41·15-s + (−0.133 − 0.232i)16-s + (0.965 − 1.67i)17-s + (−0.258 + 0.448i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151751517\)
\(L(\frac12)\) \(\approx\) \(1.151751517\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - iT \)
37 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.93T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.23051374296254135981624712985, −9.522966512771393580667117193787, −8.985797390873497029928912118020, −7.58788551578528918383432992947, −6.93271988197264275685914102626, −5.99378866583049362706151606247, −5.46773350018097362930187931929, −3.09679861644815985032244113050, −2.65350285903769048087913627467, −1.62370348046287984118941443287, 1.84121994387762301705116869059, 3.45122228172257330842942114464, 4.18248824110264974680658366725, 5.34198284131863176656115781776, 6.20443898675374747525096835912, 7.58491781773569458702825785501, 8.113945034163862006341723006323, 8.980110500624422948341358206999, 9.574952104863658833803673971764, 10.49649093462412030150708824970

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