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Properties

Label 2-777-777.317-c0-0-0
Degree $2$
Conductor $777$
Sign $-0.0715 - 0.997i$
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  + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s − 7-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)12-s + (−0.439 + 0.524i)13-s + (−0.939 − 0.342i)16-s + (1.70 + 0.300i)19-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)25-s + (−0.500 + 0.866i)27-s + (0.173 − 0.984i)28-s + (1.11 − 0.642i)31-s − 36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s − 7-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)12-s + (−0.439 + 0.524i)13-s + (−0.939 − 0.342i)16-s + (1.70 + 0.300i)19-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)25-s + (−0.500 + 0.866i)27-s + (0.173 − 0.984i)28-s + (1.11 − 0.642i)31-s − 36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0715 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0715 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.0715 - 0.997i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ -0.0715 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056131034\)
\(L(\frac12)\) \(\approx\) \(1.056131034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + 1.28iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.44849061402534068731222599909, −9.689381596106315366778409962220, −9.153718241501313923229902603208, −8.213903159195545650720077445035, −7.49188429768489594036360568426, −6.57241844876253723168758721947, −5.14864432431577108669400741791, −4.11267278705551793109626169645, −3.31843628793299285280913607217, −2.47689531697911232153191248352, 1.11055239638470763042063006011, 2.62415752924197355031849009119, 3.55886064988545938686045605812, 5.04348258416344247555442391760, 5.97639718408073298180392675776, 6.86880417491188382892843163543, 7.58984005310315589619692634707, 8.743996929663877206125220366964, 9.612391743797165691279661365979, 9.871202278927058212800892000089

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