Properties

Label 2-772-772.131-c0-0-0
Degree $2$
Conductor $772$
Sign $0.980 + 0.196i$
Analytic cond. $0.385278$
Root an. cond. $0.620707$
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  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−1.05 − 0.357i)5-s + (0.382 − 0.923i)8-s i·9-s + (0.491 − 0.996i)10-s + (0.996 − 1.49i)13-s + (0.866 + 0.5i)16-s + (0.293 + 0.257i)17-s + (0.991 + 0.130i)18-s + (0.923 + 0.617i)20-s + (0.186 + 0.142i)25-s + (1.34 + 1.18i)26-s + (−0.735 + 0.491i)29-s + (−0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−1.05 − 0.357i)5-s + (0.382 − 0.923i)8-s i·9-s + (0.491 − 0.996i)10-s + (0.996 − 1.49i)13-s + (0.866 + 0.5i)16-s + (0.293 + 0.257i)17-s + (0.991 + 0.130i)18-s + (0.923 + 0.617i)20-s + (0.186 + 0.142i)25-s + (1.34 + 1.18i)26-s + (−0.735 + 0.491i)29-s + (−0.608 + 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $0.980 + 0.196i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ 0.980 + 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6512849801\)
\(L(\frac12)\) \(\approx\) \(0.6512849801\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 - 0.991i)T \)
193 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + iT^{2} \)
5 \( 1 + (1.05 + 0.357i)T + (0.793 + 0.608i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.996 + 1.49i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (-0.293 - 0.257i)T + (0.130 + 0.991i)T^{2} \)
19 \( 1 + (-0.793 - 0.608i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.735 - 0.491i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (-0.0420 + 0.641i)T + (-0.991 - 0.130i)T^{2} \)
41 \( 1 + (-1.31 + 1.50i)T + (-0.130 - 0.991i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.991 - 0.130i)T^{2} \)
53 \( 1 + (1.31 + 0.0862i)T + (0.991 + 0.130i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.123 - 0.0420i)T + (0.793 - 0.608i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.0983 - 1.50i)T + (-0.991 + 0.130i)T^{2} \)
79 \( 1 + (0.793 - 0.608i)T^{2} \)
83 \( 1 + (0.258 + 0.965i)T^{2} \)
89 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
97 \( 1 + (-0.0675 - 0.513i)T + (-0.965 + 0.258i)T^{2} \)
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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

−10.39622063748128247404959617435, −9.343503348777882626738601443652, −8.583282472477595863386537940622, −7.932792253897091638292372464585, −7.17415319831907025320090946741, −6.07210579001621675396537752089, −5.37441523616598936186649318537, −4.03539059879277412879865293447, −3.48257565512530473498947358051, −0.792217138681178982106963346258, 1.68730861585721120479540025380, 3.00953623541939415058228834440, 4.05437645517884416856738594665, 4.70575439406806403146985998562, 6.10638313214115287858329920208, 7.45280077415105426897683481150, 8.026051907384137325854185045069, 8.965400311620495043852976280336, 9.771750640811795262566272635500, 10.90536022719827905948197136823

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