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Properties

Label 2-765-85.14-c0-0-1
Degree $2$
Conductor $765$
Sign $0.861 + 0.507i$
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  + (0.149 + 0.360i)2-s + (0.599 − 0.599i)4-s + (−0.555 − 0.831i)5-s + (0.666 + 0.275i)8-s + (0.216 − 0.324i)10-s − 0.566i·16-s + (−0.980 − 0.195i)17-s + (0.707 − 0.292i)19-s + (−0.831 − 0.165i)20-s + (1.38 − 0.275i)23-s + (−0.382 + 0.923i)25-s + (−1.08 − 0.216i)31-s + (0.870 − 0.360i)32-s + (−0.0761 − 0.382i)34-s + (0.211 + 0.211i)38-s + ⋯
L(s)  = 1  + (0.149 + 0.360i)2-s + (0.599 − 0.599i)4-s + (−0.555 − 0.831i)5-s + (0.666 + 0.275i)8-s + (0.216 − 0.324i)10-s − 0.566i·16-s + (−0.980 − 0.195i)17-s + (0.707 − 0.292i)19-s + (−0.831 − 0.165i)20-s + (1.38 − 0.275i)23-s + (−0.382 + 0.923i)25-s + (−1.08 − 0.216i)31-s + (0.870 − 0.360i)32-s + (−0.0761 − 0.382i)34-s + (0.211 + 0.211i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(0.381784\)
Root analytic conductor: \(0.617887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :0),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110294647\)
\(L(\frac12)\) \(\approx\) \(1.110294647\)
\(L(1)\) 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 \( 1 + (0.555 + 0.831i)T \)
17 \( 1 + (0.980 + 0.195i)T \)
good2 \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
53 \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.382 - 0.923i)T^{2} \)
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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

−10.66977553297740007510348670175, −9.412119494162438673174950567117, −8.856305725151448354048793704220, −7.64934920828001870299166952659, −7.09373343240269684235745892691, −6.00256515611371438780367055279, −5.08978721103483715784881114472, −4.35044469541934666909363274080, −2.82012841196487127354143360108, −1.29638118279247382254072612825, 2.01592566589998546916894904115, 3.17451362025074767724300834827, 3.83110626620090152401978094844, 5.11671105743988763211051221270, 6.56849147095198230860365947694, 7.07542333782128408651290389316, 7.902436304625418317624489987837, 8.817059504987637335439445258714, 10.00596416406809220107507156621, 10.88137661778045858468578817577

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