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Properties

Label 2-777-777.158-c0-0-0
Degree $2$
Conductor $777$
Sign $0.425 - 0.904i$
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  + i·3-s + 4-s + (0.866 + 0.5i)5-s − 7-s − 9-s + (0.866 + 0.5i)11-s + i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + 16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)20-s − i·21-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + i·3-s + 4-s + (0.866 + 0.5i)5-s − 7-s − 9-s + (0.866 + 0.5i)11-s + i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + 16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)20-s − i·21-s + (−0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ 0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212808711\)
\(L(\frac12)\) \(\approx\) \(1.212808711\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + T \)
37 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−10.43180022472442512098251705907, −9.870918594307933278462353035157, −9.430311670465312924872780039620, −8.133377617388657497031285231344, −6.92066712265916453693601821692, −6.26176059763080605983809641583, −5.57332785068276997031796581400, −4.15760415999655718854245728554, −3.05491057066444790022857253562, −2.24965620941129184699631826439, 1.50771856147652828742694032624, 2.41266520564424045586557182043, 3.61627435696262122129021532020, 5.40108501579479997664772065949, 6.24450487259573893895048261507, 6.71036483436032288389613866602, 7.51734788629130145078975354565, 8.780700043928270070380000832686, 9.385986655692561758725758469378, 10.34410250834405517083825140894

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