Properties

Label 2-765-85.24-c0-0-0
Degree $2$
Conductor $765$
Sign $0.986 - 0.163i$
Analytic cond. $0.381784$
Root an. cond. $0.617887$
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  + (−1.02 − 0.425i)2-s + (0.165 + 0.165i)4-s + (−0.980 + 0.195i)5-s + (0.325 + 0.785i)8-s + (1.08 + 0.216i)10-s − 1.17i·16-s + (0.831 + 0.555i)17-s + (−0.707 + 1.70i)19-s + (−0.195 − 0.130i)20-s + (1.17 − 0.785i)23-s + (0.923 − 0.382i)25-s + (1.63 + 1.08i)31-s + (−0.176 + 0.425i)32-s + (−0.617 − 0.923i)34-s + (1.45 − 1.45i)38-s + ⋯
L(s)  = 1  + (−1.02 − 0.425i)2-s + (0.165 + 0.165i)4-s + (−0.980 + 0.195i)5-s + (0.325 + 0.785i)8-s + (1.08 + 0.216i)10-s − 1.17i·16-s + (0.831 + 0.555i)17-s + (−0.707 + 1.70i)19-s + (−0.195 − 0.130i)20-s + (1.17 − 0.785i)23-s + (0.923 − 0.382i)25-s + (1.63 + 1.08i)31-s + (−0.176 + 0.425i)32-s + (−0.617 − 0.923i)34-s + (1.45 − 1.45i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(0.381784\)
Root analytic conductor: \(0.617887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :0),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4466740220\)
\(L(\frac12)\) \(\approx\) \(0.4466740220\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.980 - 0.195i)T \)
17 \( 1 + (-0.831 - 0.555i)T \)
good2 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
53 \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.923 + 0.382i)T^{2} \)
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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

−10.46018197848925540988559831921, −9.903191108964573339597112424900, −8.635290073365943857985782644452, −8.299200808184189317725413090119, −7.45216382100004641348242917006, −6.34785120656922232235895522749, −5.09471889277229155473594998411, −4.01647719670322119388976572929, −2.81836618054093303435389349861, −1.25131028205294972961245863721, 0.816155180189329016130218229162, 2.94710363602184265221011741971, 4.15002863137145274415409051657, 5.05507952016049189065111677994, 6.57299096343413816175048736469, 7.29939784073281190307430901899, 7.965007647169223652989169923320, 8.790876703277700439977616815864, 9.385960109557272621761689372229, 10.33882411427812998442888800156

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