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Properties

Label 2-765-85.64-c1-0-6
Degree $2$
Conductor $765$
Sign $0.607 - 0.794i$
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  + 0.232·2-s − 1.94·4-s + (−2.04 − 0.898i)5-s + (−1.46 − 1.46i)7-s − 0.917·8-s + (−0.475 − 0.208i)10-s + (0.339 − 0.339i)11-s + 4.07i·13-s + (−0.339 − 0.339i)14-s + 3.67·16-s + (3.75 − 1.69i)17-s + 4i·19-s + (3.98 + 1.74i)20-s + (0.0788 − 0.0788i)22-s + (5.76 + 5.76i)23-s + ⋯
L(s)  = 1  + 0.164·2-s − 0.972·4-s + (−0.915 − 0.401i)5-s + (−0.552 − 0.552i)7-s − 0.324·8-s + (−0.150 − 0.0660i)10-s + (0.102 − 0.102i)11-s + 1.12i·13-s + (−0.0907 − 0.0907i)14-s + 0.919·16-s + (0.911 − 0.410i)17-s + 0.917i·19-s + (0.891 + 0.390i)20-s + (0.0168 − 0.0168i)22-s + (1.20 + 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724634 + 0.357946i\)
\(L(\frac12)\) \(\approx\) \(0.724634 + 0.357946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 + 0.898i)T \)
17 \( 1 + (-3.75 + 1.69i)T \)
good2 \( 1 - 0.232T + 2T^{2} \)
7 \( 1 + (1.46 + 1.46i)T + 7iT^{2} \)
11 \( 1 + (-0.339 + 0.339i)T - 11iT^{2} \)
13 \( 1 - 4.07iT - 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-5.76 - 5.76i)T + 23iT^{2} \)
29 \( 1 + (0.732 + 0.732i)T + 29iT^{2} \)
31 \( 1 + (4.28 + 4.28i)T + 31iT^{2} \)
37 \( 1 + (-0.917 + 0.917i)T - 37iT^{2} \)
41 \( 1 + (7.62 - 7.62i)T - 41iT^{2} \)
43 \( 1 - 7.45T + 43T^{2} \)
47 \( 1 - 3.60iT - 47T^{2} \)
53 \( 1 - 6.14T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (4 - 4i)T - 61iT^{2} \)
67 \( 1 - 3.14iT - 67T^{2} \)
71 \( 1 + (1.28 + 1.28i)T + 71iT^{2} \)
73 \( 1 + (8.60 - 8.60i)T - 73iT^{2} \)
79 \( 1 + (-7.23 + 7.23i)T - 79iT^{2} \)
83 \( 1 - 2.23T + 83T^{2} \)
89 \( 1 - 9.37T + 89T^{2} \)
97 \( 1 + (-11.8 + 11.8i)T - 97iT^{2} \)
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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

−10.31239634157753237174309823132, −9.429419355293561911431911521898, −8.918113907410682078390419636505, −7.81294850829956257616368989307, −7.20204204076868608064056265355, −5.88453109444025315081882493050, −4.88847322620778648717832215712, −3.96274997932251519919968463927, −3.36371808924042593337210169734, −1.08730704033042520704374628251, 0.51129865419670296034425426750, 2.92069822875406816432002433695, 3.59230682152183154950094307132, 4.78033347189367132984270562449, 5.58800008722968150093480647953, 6.74196647877672722293638843492, 7.70532835170448388113405443862, 8.595461982017109714244583777262, 9.174937277298127336347124142679, 10.31195839288421472458043884424

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