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Properties

Label 2-765-17.8-c1-0-14
Degree $2$
Conductor $765$
Sign $0.960 + 0.276i$
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  + (−1.09 + 1.09i)2-s − 0.403i·4-s + (0.923 + 0.382i)5-s + (−3.45 + 1.43i)7-s + (−1.74 − 1.74i)8-s + (−1.43 + 0.593i)10-s + (−0.558 − 1.34i)11-s − 6.71i·13-s + (2.21 − 5.35i)14-s + 4.64·16-s + (4.12 − 0.148i)17-s + (−1.32 + 1.32i)19-s + (0.154 − 0.373i)20-s + (2.08 + 0.865i)22-s + (−1.61 − 3.90i)23-s + ⋯
L(s)  = 1  + (−0.775 + 0.775i)2-s − 0.201i·4-s + (0.413 + 0.171i)5-s + (−1.30 + 0.541i)7-s + (−0.618 − 0.618i)8-s + (−0.452 + 0.187i)10-s + (−0.168 − 0.406i)11-s − 1.86i·13-s + (0.593 − 1.43i)14-s + 1.16·16-s + (0.999 − 0.0360i)17-s + (−0.304 + 0.304i)19-s + (0.0345 − 0.0834i)20-s + (0.445 + 0.184i)22-s + (−0.337 − 0.815i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.960 + 0.276i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (586, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.960 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650150 - 0.0918375i\)
\(L(\frac12)\) \(\approx\) \(0.650150 - 0.0918375i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-4.12 + 0.148i)T \)
good2 \( 1 + (1.09 - 1.09i)T - 2iT^{2} \)
7 \( 1 + (3.45 - 1.43i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.558 + 1.34i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.71iT - 13T^{2} \)
19 \( 1 + (1.32 - 1.32i)T - 19iT^{2} \)
23 \( 1 + (1.61 + 3.90i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-7.01 - 2.90i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.495 + 1.19i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.72 + 4.17i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.45 - 0.601i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.56 + 4.56i)T + 43iT^{2} \)
47 \( 1 + 2.36iT - 47T^{2} \)
53 \( 1 + (4.25 - 4.25i)T - 53iT^{2} \)
59 \( 1 + (7.29 + 7.29i)T + 59iT^{2} \)
61 \( 1 + (-2.90 + 1.20i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 + (-1.46 + 3.53i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-12.9 - 5.35i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.916 + 2.21i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 - 1.59iT - 89T^{2} \)
97 \( 1 + (12.5 + 5.20i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.06359496089401695757452029168, −9.418552268737732527846980152493, −8.409435504253283718178380505451, −7.912829451974130327299036414482, −6.75131874131506150678445071062, −6.09707135183683170718733930080, −5.39016575334905053997463562521, −3.44603156077197514282307890404, −2.83597991554830922147614550886, −0.47157434790808523950153691320, 1.24861474711195062592312538988, 2.45874493066201243930403409808, 3.62112240087028161531879768818, 4.87917323506633070366890832113, 6.20022893942732090535709000376, 6.77009270529994416339064224237, 8.029197255494417138622123932998, 9.081280649434306963163166861246, 9.790968895730249881624984264507, 9.965403765476314750268859390402

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