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Properties

Label 2-765-85.44-c0-0-0
Degree $2$
Conductor $765$
Sign $0.855 - 0.518i$
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  + (−1.53 − 0.636i)2-s + (1.24 + 1.24i)4-s + (0.195 + 0.980i)5-s + (−0.487 − 1.17i)8-s + (0.324 − 1.63i)10-s + 0.351i·16-s + (0.555 − 0.831i)17-s + (−0.707 + 1.70i)19-s + (−0.980 + 1.46i)20-s + (0.785 + 1.17i)23-s + (−0.923 + 0.382i)25-s + (−0.216 + 0.324i)31-s + (−0.263 + 0.636i)32-s + (−1.38 + 0.923i)34-s + (2.17 − 2.17i)38-s + ⋯
L(s)  = 1  + (−1.53 − 0.636i)2-s + (1.24 + 1.24i)4-s + (0.195 + 0.980i)5-s + (−0.487 − 1.17i)8-s + (0.324 − 1.63i)10-s + 0.351i·16-s + (0.555 − 0.831i)17-s + (−0.707 + 1.70i)19-s + (−0.980 + 1.46i)20-s + (0.785 + 1.17i)23-s + (−0.923 + 0.382i)25-s + (−0.216 + 0.324i)31-s + (−0.263 + 0.636i)32-s + (−1.38 + 0.923i)34-s + (2.17 − 2.17i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.855 - 0.518i$
Analytic conductor: \(0.381784\)
Root analytic conductor: \(0.617887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :0),\ 0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4616208575\)
\(L(\frac12)\) \(\approx\) \(0.4616208575\)
\(L(1)\) 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 \( 1 + (-0.195 - 0.980i)T \)
17 \( 1 + (-0.555 + 0.831i)T \)
good2 \( 1 + (1.53 + 0.636i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.785 - 1.17i)T + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
53 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.923 + 0.382i)T^{2} \)
79 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2} \)
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

−10.56534328404653758921317930238, −9.722759403783153339631725197834, −9.231609120453851546204736914388, −8.027761905700390712106140041187, −7.52381397399484372109789459904, −6.58872321990610938161778641589, −5.45904000963447356297657014896, −3.66579801135561116035596686590, −2.72097161958540813175416436488, −1.55814090834279787235449273614, 0.862381926309192441169465725467, 2.28369476668750028460570189517, 4.22585141818939716307997360605, 5.36670013829349201144436361482, 6.35886503379076851345358845185, 7.17944991556149346937566088782, 8.141250373250028625507327737846, 8.847308053642750628645363121411, 9.212414702478631512199412871604, 10.31225717059002539488056265329

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