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Properties

Label 2-765-85.64-c1-0-6
Degree 22
Conductor 765765
Sign 0.607−0.794i0.607 - 0.794i
Analytic cond. 6.108556.10855
Root an. cond. 2.471542.47154
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(765s/2 ΓC(s) L(s)=((0.607−0.794i) Λ‾(2−s)\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}
Λ(s)=(765s/2 ΓC(s+1/2) L(s)=((0.607−0.794i) Λ‾(1−s)\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: 22
Conductor: 765765    =    32⋅5⋅173^{2} \cdot 5 \cdot 17
Sign: 0.607−0.794i0.607 - 0.794i
Analytic conductor: 6.108556.10855
Root analytic conductor: 2.471542.47154
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ765(64,⋅)\chi_{765} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 765, ( :1/2), 0.607−0.794i)(2,\ 765,\ (\ :1/2),\ 0.607 - 0.794i)

Particular Values

L(1)L(1) ≈\approx 0.724634+0.357946i0.724634 + 0.357946i
L(12)L(\frac12) ≈\approx 0.724634+0.357946i0.724634 + 0.357946i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1+(2.04+0.898i)T 1 + (2.04 + 0.898i)T
17 1+(−3.75+1.69i)T 1 + (-3.75 + 1.69i)T
good2 1−0.232T+2T2 1 - 0.232T + 2T^{2}
7 1+(1.46+1.46i)T+7iT2 1 + (1.46 + 1.46i)T + 7iT^{2}
11 1+(−0.339+0.339i)T−11iT2 1 + (-0.339 + 0.339i)T - 11iT^{2}
13 1−4.07iT−13T2 1 - 4.07iT - 13T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+(−5.76−5.76i)T+23iT2 1 + (-5.76 - 5.76i)T + 23iT^{2}
29 1+(0.732+0.732i)T+29iT2 1 + (0.732 + 0.732i)T + 29iT^{2}
31 1+(4.28+4.28i)T+31iT2 1 + (4.28 + 4.28i)T + 31iT^{2}
37 1+(−0.917+0.917i)T−37iT2 1 + (-0.917 + 0.917i)T - 37iT^{2}
41 1+(7.62−7.62i)T−41iT2 1 + (7.62 - 7.62i)T - 41iT^{2}
43 1−7.45T+43T2 1 - 7.45T + 43T^{2}
47 1−3.60iT−47T2 1 - 3.60iT - 47T^{2}
53 1−6.14T+53T2 1 - 6.14T + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1+(4−4i)T−61iT2 1 + (4 - 4i)T - 61iT^{2}
67 1−3.14iT−67T2 1 - 3.14iT - 67T^{2}
71 1+(1.28+1.28i)T+71iT2 1 + (1.28 + 1.28i)T + 71iT^{2}
73 1+(8.60−8.60i)T−73iT2 1 + (8.60 - 8.60i)T - 73iT^{2}
79 1+(−7.23+7.23i)T−79iT2 1 + (-7.23 + 7.23i)T - 79iT^{2}
83 1−2.23T+83T2 1 - 2.23T + 83T^{2}
89 1−9.37T+89T2 1 - 9.37T + 89T^{2}
97 1+(−11.8+11.8i)T−97iT2 1 + (-11.8 + 11.8i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line