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Properties

Label 2-819-7.2-c1-0-28
Degree 22
Conductor 819819
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 6.539746.53974
Root an. cond. 2.557292.55729
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.330i)2-s + (0.927 + 1.60i)4-s + (−1.11 + 1.93i)5-s + (−2 − 1.73i)7-s − 1.47·8-s + (−0.427 − 0.739i)10-s + (−1.5 − 2.59i)11-s − 13-s + (0.954 − 0.330i)14-s + (−1.57 + 2.72i)16-s + (−3.73 − 6.47i)17-s + (−1.5 + 2.59i)19-s − 4.14·20-s + 1.14·22-s + (−1.88 + 3.25i)23-s + ⋯
L(s)  = 1  + (−0.135 + 0.233i)2-s + (0.463 + 0.802i)4-s + (−0.499 + 0.866i)5-s + (−0.755 − 0.654i)7-s − 0.520·8-s + (−0.135 − 0.233i)10-s + (−0.452 − 0.783i)11-s − 0.277·13-s + (0.255 − 0.0884i)14-s + (−0.393 + 0.681i)16-s + (−0.906 − 1.56i)17-s + (−0.344 + 0.596i)19-s − 0.927·20-s + 0.244·22-s + (−0.392 + 0.679i)23-s + ⋯

Functional equation

Λ(s)=(819s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(819s/2 ΓC(s+1/2) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 819819    =    32⋅7⋅133^{2} \cdot 7 \cdot 13
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 6.539746.53974
Root analytic conductor: 2.557292.55729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ819(352,⋅)\chi_{819} (352, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 819, ( :1/2), −0.605+0.795i)(2,\ 819,\ (\ :1/2),\ -0.605 + 0.795i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7 1+(2+1.73i)T 1 + (2 + 1.73i)T
13 1+T 1 + T
good2 1+(0.190−0.330i)T+(−1−1.73i)T2 1 + (0.190 - 0.330i)T + (-1 - 1.73i)T^{2}
5 1+(1.11−1.93i)T+(−2.5−4.33i)T2 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.5+2.59i)T+(−5.5+9.52i)T2 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.73+6.47i)T+(−8.5+14.7i)T2 1 + (3.73 + 6.47i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.5−2.59i)T+(−9.5−16.4i)T2 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.88−3.25i)T+(−11.5−19.9i)T2 1 + (1.88 - 3.25i)T + (-11.5 - 19.9i)T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1+(2.5+4.33i)T+(−15.5+26.8i)T2 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2}
37 1+(−4.35+7.54i)T+(−18.5−32.0i)T2 1 + (-4.35 + 7.54i)T + (-18.5 - 32.0i)T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1+(−0.736+1.27i)T+(−23.5−40.7i)T2 1 + (-0.736 + 1.27i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.736−1.27i)T+(−26.5+45.8i)T2 1 + (-0.736 - 1.27i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.73−6.47i)T+(−29.5+51.0i)T2 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.5−2.59i)T+(−30.5−52.8i)T2 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.5−2.59i)T+(−33.5+58.0i)T2 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1+(5.35+9.27i)T+(−36.5+63.2i)T2 1 + (5.35 + 9.27i)T + (-36.5 + 63.2i)T^{2}
79 1+(5.35−9.27i)T+(−39.5−68.4i)T2 1 + (5.35 - 9.27i)T + (-39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(1.11−1.93i)T+(−44.5−77.0i)T2 1 + (1.11 - 1.93i)T + (-44.5 - 77.0i)T^{2}
97 1+17.4T+97T2 1 + 17.4T + 97T^{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

−9.987500996577917549745758689212, −9.010846694561937209699595201593, −8.005039148951372656891182649583, −7.24395070788793534298071109941, −6.80525128809619528921993459716, −5.77584626204260266496645621454, −4.20332010091474315354451546956, −3.29793360871651540590775749793, −2.58352746159938435446194787789, 0, 1.74653484422290677461957236649, 2.82323335618506260299832368688, 4.37097026767864617643545308797, 5.14878668267496541480930348675, 6.29066252686543234164784665600, 6.84514663199993461863276043130, 8.312429183163123294626687571934, 8.785139780967327474852124639337, 9.864856940100742380157567092400

Graph of the ZZ-function along the critical line