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Properties

Label 2-1134-21.5-c1-0-29
Degree 22
Conductor 11341134
Sign −0.694+0.719i-0.694 + 0.719i
Analytic cond. 9.055039.05503
Root an. cond. 3.009153.00915
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.82 − 3.15i)5-s + (−1.04 − 2.43i)7-s − 0.999i·8-s + (−3.15 + 1.82i)10-s + (4.38 − 2.53i)11-s − 3.39i·13-s + (−0.310 + 2.62i)14-s + (−0.5 + 0.866i)16-s + (0.774 + 1.34i)17-s + (−0.707 − 0.408i)19-s + 3.64·20-s − 5.06·22-s + (1.47 + 0.850i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.814 − 1.41i)5-s + (−0.394 − 0.918i)7-s − 0.353i·8-s + (−0.997 + 0.576i)10-s + (1.32 − 0.763i)11-s − 0.942i·13-s + (−0.0829 + 0.702i)14-s + (−0.125 + 0.216i)16-s + (0.187 + 0.325i)17-s + (−0.162 − 0.0936i)19-s + 0.814·20-s − 1.08·22-s + (0.307 + 0.177i)23-s + ⋯

Functional equation

Λ(s)=(1134s/2 ΓC(s) L(s)=((−0.694+0.719i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1134s/2 ΓC(s+1/2) L(s)=((−0.694+0.719i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11341134    =    2⋅34⋅72 \cdot 3^{4} \cdot 7
Sign: −0.694+0.719i-0.694 + 0.719i
Analytic conductor: 9.055039.05503
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1134(971,⋅)\chi_{1134} (971, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1134, ( :1/2), −0.694+0.719i)(2,\ 1134,\ (\ :1/2),\ -0.694 + 0.719i)

Particular Values

L(1)L(1) ≈\approx 1.3636469481.363646948
L(12)L(\frac12) ≈\approx 1.3636469481.363646948
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1 1
7 1+(1.04+2.43i)T 1 + (1.04 + 2.43i)T
good5 1+(−1.82+3.15i)T+(−2.5−4.33i)T2 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2}
11 1+(−4.38+2.53i)T+(5.5−9.52i)T2 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2}
13 1+3.39iT−13T2 1 + 3.39iT - 13T^{2}
17 1+(−0.774−1.34i)T+(−8.5+14.7i)T2 1 + (-0.774 - 1.34i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.707+0.408i)T+(9.5+16.4i)T2 1 + (0.707 + 0.408i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.47−0.850i)T+(11.5+19.9i)T2 1 + (-1.47 - 0.850i)T + (11.5 + 19.9i)T^{2}
29 1−4.16iT−29T2 1 - 4.16iT - 29T^{2}
31 1+(−1.87+1.08i)T+(15.5−26.8i)T2 1 + (-1.87 + 1.08i)T + (15.5 - 26.8i)T^{2}
37 1+(3.39−5.88i)T+(−18.5−32.0i)T2 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2}
41 1−2.03T+41T2 1 - 2.03T + 41T^{2}
43 1+6.12T+43T2 1 + 6.12T + 43T^{2}
47 1+(3.37−5.83i)T+(−23.5−40.7i)T2 1 + (3.37 - 5.83i)T + (-23.5 - 40.7i)T^{2}
53 1+(−11.4+6.63i)T+(26.5−45.8i)T2 1 + (-11.4 + 6.63i)T + (26.5 - 45.8i)T^{2}
59 1+(1.08+1.88i)T+(−29.5+51.0i)T2 1 + (1.08 + 1.88i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.28−3.62i)T+(30.5+52.8i)T2 1 + (-6.28 - 3.62i)T + (30.5 + 52.8i)T^{2}
67 1+(1.22+2.12i)T+(−33.5+58.0i)T2 1 + (1.22 + 2.12i)T + (-33.5 + 58.0i)T^{2}
71 1−6.74iT−71T2 1 - 6.74iT - 71T^{2}
73 1+(−3.76+2.17i)T+(36.5−63.2i)T2 1 + (-3.76 + 2.17i)T + (36.5 - 63.2i)T^{2}
79 1+(6.37−11.0i)T+(−39.5−68.4i)T2 1 + (6.37 - 11.0i)T + (-39.5 - 68.4i)T^{2}
83 1−1.53T+83T2 1 - 1.53T + 83T^{2}
89 1+(−6.01+10.4i)T+(−44.5−77.0i)T2 1 + (-6.01 + 10.4i)T + (-44.5 - 77.0i)T^{2}
97 1+6.46iT−97T2 1 + 6.46iT - 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

−9.603164021001687318602312505094, −8.679279410884712435826126873305, −8.293609386828978736802009281872, −7.03963797875601418683356346067, −6.18755698841218423897451651772, −5.24342137824587487373454210089, −4.13820023230270475458365156611, −3.18479237834535183246706941090, −1.50335378199694477909762918554, −0.78649522100574018455393474614, 1.81283363127403922384824665575, 2.58443019615818561970158499760, 3.86616230121560733795598258440, 5.30677595084518369598400007395, 6.37874359774594553710056730822, 6.61802865327963926493607181051, 7.41199054308508908577839349789, 8.750147326903297542135259971849, 9.370790483215377024624496689780, 9.904481863556233706609255598190

Graph of the ZZ-function along the critical line