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Properties

Label 2-765-17.9-c1-0-16
Degree $2$
Conductor $765$
Sign $0.913 + 0.407i$
Analytic cond. $6.10855$
Root an. cond. $2.47154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 1.27i)2-s − 1.26i·4-s + (0.382 − 0.923i)5-s + (1.66 + 4.01i)7-s + (0.943 + 0.943i)8-s + (−0.691 − 1.66i)10-s + (−0.0485 + 0.0200i)11-s + 3.02i·13-s + (7.24 + 3.00i)14-s + 4.93·16-s + (−3.12 + 2.69i)17-s + (5.52 − 5.52i)19-s + (−1.16 − 0.482i)20-s + (−0.0362 + 0.0875i)22-s + (−0.962 + 0.398i)23-s + ⋯
L(s)  = 1  + (0.902 − 0.902i)2-s − 0.630i·4-s + (0.171 − 0.413i)5-s + (0.628 + 1.51i)7-s + (0.333 + 0.333i)8-s + (−0.218 − 0.527i)10-s + (−0.0146 + 0.00605i)11-s + 0.839i·13-s + (1.93 + 0.801i)14-s + 1.23·16-s + (−0.757 + 0.653i)17-s + (1.26 − 1.26i)19-s + (−0.260 − 0.107i)20-s + (−0.00773 + 0.0186i)22-s + (−0.200 + 0.0831i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.913 + 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70267 - 0.575754i\)
\(L(\frac12)\) \(\approx\) \(2.70267 - 0.575754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (3.12 - 2.69i)T \)
good2 \( 1 + (-1.27 + 1.27i)T - 2iT^{2} \)
7 \( 1 + (-1.66 - 4.01i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.0485 - 0.0200i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.02iT - 13T^{2} \)
19 \( 1 + (-5.52 + 5.52i)T - 19iT^{2} \)
23 \( 1 + (0.962 - 0.398i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.161 + 0.388i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.27 + 0.529i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.311 - 0.128i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.52 + 6.09i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.06 + 7.06i)T + 43iT^{2} \)
47 \( 1 - 6.13iT - 47T^{2} \)
53 \( 1 + (-8.52 + 8.52i)T - 53iT^{2} \)
59 \( 1 + (3.60 + 3.60i)T + 59iT^{2} \)
61 \( 1 + (2.28 + 5.51i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 0.916T + 67T^{2} \)
71 \( 1 + (3.86 + 1.59i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.06 + 4.98i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.22 + 3.82i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.61 + 4.61i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (7.35 - 17.7i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.58963598109610054340939384726, −9.327330733699862120890506679515, −8.796223512519852326885432486005, −7.84110853244713972531429068406, −6.50447531781236415257377032645, −5.30424648809350774742852348309, −4.93469110007269381966456054643, −3.75627067279666184005657246975, −2.49568217546019416593002744102, −1.77412452340803573666885169075, 1.24551781802767931499145740629, 3.23658836769653436909380305093, 4.17557613436474437317917393358, 5.04054079261778528165365679611, 5.91587285484701730928285604372, 6.94898744951107575234833539632, 7.48547711608312602005534249298, 8.201953071760711120354996464303, 9.812243755223993170704383514203, 10.33191663385161970034970285511

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