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Properties

Label 2-2736-19.7-c1-0-40
Degree 22
Conductor 27362736
Sign −0.856+0.516i-0.856 + 0.516i
Analytic cond. 21.847021.8470
Root an. cond. 4.674084.67408
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  + (−1.65 − 2.85i)5-s − 1.44·7-s + 1.81·11-s + (0.5 − 0.866i)13-s + (3.30 + 5.71i)17-s + (−1 − 4.24i)19-s + (2.39 − 4.14i)23-s + (−2.94 + 5.10i)25-s + (4.78 − 8.28i)29-s + 4.55·31-s + (2.39 + 4.14i)35-s − 5.89·37-s + (1.48 + 2.57i)41-s + (−4.17 − 7.22i)43-s + (1.48 − 2.57i)47-s + ⋯
L(s)  = 1  + (−0.738 − 1.27i)5-s − 0.547·7-s + 0.547·11-s + (0.138 − 0.240i)13-s + (0.800 + 1.38i)17-s + (−0.229 − 0.973i)19-s + (0.498 − 0.864i)23-s + (−0.589 + 1.02i)25-s + (0.888 − 1.53i)29-s + 0.817·31-s + (0.404 + 0.700i)35-s − 0.969·37-s + (0.231 + 0.401i)41-s + (−0.636 − 1.10i)43-s + (0.216 − 0.374i)47-s + ⋯

Functional equation

Λ(s)=(2736s/2 ΓC(s) L(s)=((−0.856+0.516i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2736s/2 ΓC(s+1/2) L(s)=((−0.856+0.516i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27362736    =    24⋅32⋅192^{4} \cdot 3^{2} \cdot 19
Sign: −0.856+0.516i-0.856 + 0.516i
Analytic conductor: 21.847021.8470
Root analytic conductor: 4.674084.67408
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2736(577,⋅)\chi_{2736} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2736, ( :1/2), −0.856+0.516i)(2,\ 2736,\ (\ :1/2),\ -0.856 + 0.516i)

Particular Values

L(1)L(1) ≈\approx 0.98162638940.9816263894
L(12)L(\frac12) ≈\approx 0.98162638940.9816263894
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)
bad2 1 1
3 1 1
19 1+(1+4.24i)T 1 + (1 + 4.24i)T
good5 1+(1.65+2.85i)T+(−2.5+4.33i)T2 1 + (1.65 + 2.85i)T + (-2.5 + 4.33i)T^{2}
7 1+1.44T+7T2 1 + 1.44T + 7T^{2}
11 1−1.81T+11T2 1 - 1.81T + 11T^{2}
13 1+(−0.5+0.866i)T+(−6.5−11.2i)T2 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.30−5.71i)T+(−8.5+14.7i)T2 1 + (-3.30 - 5.71i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.39+4.14i)T+(−11.5−19.9i)T2 1 + (-2.39 + 4.14i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.78+8.28i)T+(−14.5−25.1i)T2 1 + (-4.78 + 8.28i)T + (-14.5 - 25.1i)T^{2}
31 1−4.55T+31T2 1 - 4.55T + 31T^{2}
37 1+5.89T+37T2 1 + 5.89T + 37T^{2}
41 1+(−1.48−2.57i)T+(−20.5+35.5i)T2 1 + (-1.48 - 2.57i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.17+7.22i)T+(−21.5+37.2i)T2 1 + (4.17 + 7.22i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.48+2.57i)T+(−23.5−40.7i)T2 1 + (-1.48 + 2.57i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.65+2.85i)T+(−26.5−45.8i)T2 1 + (-1.65 + 2.85i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.21+7.29i)T+(−29.5+51.0i)T2 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.5−4.33i)T+(−30.5−52.8i)T2 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.17−12.4i)T+(−33.5−58.0i)T2 1 + (7.17 - 12.4i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.78−8.28i)T+(−35.5+61.4i)T2 1 + (-4.78 - 8.28i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.5+4.33i)T+(−36.5+63.2i)T2 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2}
79 1+(7.17+12.4i)T+(−39.5+68.4i)T2 1 + (7.17 + 12.4i)T + (-39.5 + 68.4i)T^{2}
83 1+3.63T+83T2 1 + 3.63T + 83T^{2}
89 1+(8.25−14.2i)T+(−44.5−77.0i)T2 1 + (8.25 - 14.2i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.44+11.1i)T+(−48.5+84.0i)T2 1 + (6.44 + 11.1i)T + (-48.5 + 84.0i)T^{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

−8.431854882932722705739375320267, −8.080173568945960708362840261607, −6.93500104435601145130740430796, −6.24742106321867829497895006251, −5.32537346021819970872976274628, −4.43983849594356380067889313474, −3.89476230239202835254394120667, −2.82275564538882994472971357899, −1.37159603277123062616988382934, −0.35244167096086664316184137844, 1.37474455466796349619980555794, 3.07076986235672709601776021540, 3.16544232375494819346943649335, 4.25545554546828768554245358538, 5.29858735637684629927578550219, 6.32583406868920099758094559385, 6.90449559201036933871420981140, 7.46122664879619792107389713502, 8.261406388578560256313244329751, 9.252211836966521854377740079000

Graph of the ZZ-function along the critical line