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Properties

Label 2-777-777.443-c0-0-3
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: $\chi_{777} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ 0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.253891678\)
\(L(\frac12)\) \(\approx\) \(1.253891678\)
\(L(1)\) 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

−10.69125817450908597747122306117, −10.17983395733295444928740456673, −9.042834495357797784557216094931, −7.897558273702863549087789413698, −7.35287055426919134139449222551, −6.60786926126503001800078379752, −4.75485280971330933914854571202, −4.06018783857072221558492935730, −3.18859951709655591509421011586, −2.52945613370736458552621321082, 1.33327395190103559994330426460, 2.47623018926985748667482892209, 4.13984381737386359188479823955, 5.08602977340909211079185199943, 6.15016292497083434073664112052, 6.67611173140684066525039000659, 8.068155912710281049042149338856, 8.361242066513964661430764522362, 9.168990318881946110108840436478, 10.33508626705359493601586219272

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