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Properties

Label 2-2268-63.58-c1-0-8
Degree 22
Conductor 22682268
Sign 0.999+0.0354i0.999 + 0.0354i
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  + (−2.00 − 3.47i)5-s + (−1.89 + 1.84i)7-s + (0.885 − 1.53i)11-s + (−0.114 + 0.198i)13-s + (3.04 + 5.27i)17-s + (−3.27 + 5.67i)19-s + (0.769 + 1.33i)23-s + (−5.55 + 9.62i)25-s + (0.271 + 0.469i)29-s + 4.55·31-s + (10.2 + 2.86i)35-s + (1.54 − 2.66i)37-s + (4.43 − 7.69i)41-s + (−2.12 − 3.67i)43-s − 0.757·47-s + ⋯
L(s)  = 1  + (−0.897 − 1.55i)5-s + (−0.715 + 0.698i)7-s + (0.267 − 0.462i)11-s + (−0.0317 + 0.0549i)13-s + (0.739 + 1.28i)17-s + (−0.752 + 1.30i)19-s + (0.160 + 0.277i)23-s + (−1.11 + 1.92i)25-s + (0.0503 + 0.0872i)29-s + 0.817·31-s + (1.72 + 0.484i)35-s + (0.253 − 0.438i)37-s + (0.693 − 1.20i)41-s + (−0.323 − 0.560i)43-s − 0.110·47-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.999+0.0354i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.999+0.0354i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.999+0.0354i0.999 + 0.0354i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(2053,⋅)\chi_{2268} (2053, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.999+0.0354i)(2,\ 2268,\ (\ :1/2),\ 0.999 + 0.0354i)

Particular Values

L(1)L(1) ≈\approx 1.1738866241.173886624
L(12)L(\frac12) ≈\approx 1.1738866241.173886624
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
7 1+(1.89−1.84i)T 1 + (1.89 - 1.84i)T
good5 1+(2.00+3.47i)T+(−2.5+4.33i)T2 1 + (2.00 + 3.47i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.885+1.53i)T+(−5.5−9.52i)T2 1 + (-0.885 + 1.53i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.114−0.198i)T+(−6.5−11.2i)T2 1 + (0.114 - 0.198i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.04−5.27i)T+(−8.5+14.7i)T2 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.27−5.67i)T+(−9.5−16.4i)T2 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.769−1.33i)T+(−11.5+19.9i)T2 1 + (-0.769 - 1.33i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.271−0.469i)T+(−14.5+25.1i)T2 1 + (-0.271 - 0.469i)T + (-14.5 + 25.1i)T^{2}
31 1−4.55T+31T2 1 - 4.55T + 31T^{2}
37 1+(−1.54+2.66i)T+(−18.5−32.0i)T2 1 + (-1.54 + 2.66i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.43+7.69i)T+(−20.5−35.5i)T2 1 + (-4.43 + 7.69i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.12+3.67i)T+(−21.5+37.2i)T2 1 + (2.12 + 3.67i)T + (-21.5 + 37.2i)T^{2}
47 1+0.757T+47T2 1 + 0.757T + 47T^{2}
53 1+(3.19+5.53i)T+(−26.5+45.8i)T2 1 + (3.19 + 5.53i)T + (-26.5 + 45.8i)T^{2}
59 1−5.17T+59T2 1 - 5.17T + 59T^{2}
61 1−12.5T+61T2 1 - 12.5T + 61T^{2}
67 1+6.18T+67T2 1 + 6.18T + 67T^{2}
71 1−13.9T+71T2 1 - 13.9T + 71T^{2}
73 1+(5.08+8.81i)T+(−36.5+63.2i)T2 1 + (5.08 + 8.81i)T + (-36.5 + 63.2i)T^{2}
79 1−11.5T+79T2 1 - 11.5T + 79T^{2}
83 1+(−8.66−15.0i)T+(−41.5+71.8i)T2 1 + (-8.66 - 15.0i)T + (-41.5 + 71.8i)T^{2}
89 1+(5.04−8.73i)T+(−44.5−77.0i)T2 1 + (5.04 - 8.73i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.91−8.50i)T+(−48.5+84.0i)T2 1 + (-4.91 - 8.50i)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

−8.851413996036036358783148144834, −8.307664858054532662622399289419, −7.83318694672567656772135386298, −6.55566554329002466176940494952, −5.76176291975602524304829360125, −5.14037039234412391266273016720, −3.91796135469996504475415347449, −3.63069689372264375410001919290, −2.00587021909282220706459695606, −0.794302985389181549296926861773, 0.60625718088102149626618487091, 2.62489801372902723628624452862, 3.09494794740767019956205277902, 4.07279140431633995159834112035, 4.81988652290861301554137282913, 6.34188616838061545163851131999, 6.72629448868744916537278443093, 7.38886901844042317770446777731, 7.960656910820860828972575958614, 9.144520690992196915682417034043

Graph of the ZZ-function along the critical line