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Properties

Label 2-966-161.145-c1-0-8
Degree 22
Conductor 966966
Sign −0.900−0.435i-0.900 - 0.435i
Analytic cond. 7.713547.71354
Root an. cond. 2.777322.77732
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  + (0.327 + 0.945i)2-s + (−0.458 + 0.888i)3-s + (−0.786 + 0.618i)4-s + (1.68 + 0.161i)5-s + (−0.989 − 0.142i)6-s + (−0.672 + 2.55i)7-s + (−0.841 − 0.540i)8-s + (−0.580 − 0.814i)9-s + (0.399 + 1.64i)10-s + (3.81 + 1.31i)11-s + (−0.189 − 0.981i)12-s + (−0.831 + 2.83i)13-s + (−2.63 + 0.201i)14-s + (−0.916 + 1.42i)15-s + (0.235 − 0.971i)16-s + (2.93 − 1.17i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.668i)2-s + (−0.264 + 0.513i)3-s + (−0.393 + 0.309i)4-s + (0.754 + 0.0720i)5-s + (−0.404 − 0.0580i)6-s + (−0.254 + 0.967i)7-s + (−0.297 − 0.191i)8-s + (−0.193 − 0.271i)9-s + (0.126 + 0.521i)10-s + (1.14 + 0.397i)11-s + (−0.0546 − 0.283i)12-s + (−0.230 + 0.785i)13-s + (−0.705 + 0.0539i)14-s + (−0.236 + 0.368i)15-s + (0.0589 − 0.242i)16-s + (0.712 − 0.285i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((−0.900−0.435i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((−0.900−0.435i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: −0.900−0.435i-0.900 - 0.435i
Analytic conductor: 7.713547.71354
Root analytic conductor: 2.777322.77732
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ966(145,⋅)\chi_{966} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 966, ( :1/2), −0.900−0.435i)(2,\ 966,\ (\ :1/2),\ -0.900 - 0.435i)

Particular Values

L(1)L(1) ≈\approx 0.366092+1.59641i0.366092 + 1.59641i
L(12)L(\frac12) ≈\approx 0.366092+1.59641i0.366092 + 1.59641i
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+(−0.327−0.945i)T 1 + (-0.327 - 0.945i)T
3 1+(0.458−0.888i)T 1 + (0.458 - 0.888i)T
7 1+(0.672−2.55i)T 1 + (0.672 - 2.55i)T
23 1+(1.54−4.54i)T 1 + (1.54 - 4.54i)T
good5 1+(−1.68−0.161i)T+(4.90+0.946i)T2 1 + (-1.68 - 0.161i)T + (4.90 + 0.946i)T^{2}
11 1+(−3.81−1.31i)T+(8.64+6.79i)T2 1 + (-3.81 - 1.31i)T + (8.64 + 6.79i)T^{2}
13 1+(0.831−2.83i)T+(−10.9−7.02i)T2 1 + (0.831 - 2.83i)T + (-10.9 - 7.02i)T^{2}
17 1+(−2.93+1.17i)T+(12.3−11.7i)T2 1 + (-2.93 + 1.17i)T + (12.3 - 11.7i)T^{2}
19 1+(0.633+0.253i)T+(13.7+13.1i)T2 1 + (0.633 + 0.253i)T + (13.7 + 13.1i)T^{2}
29 1+(−0.370+2.57i)T+(−27.8−8.17i)T2 1 + (-0.370 + 2.57i)T + (-27.8 - 8.17i)T^{2}
31 1+(−1.49−0.0713i)T+(30.8+2.94i)T2 1 + (-1.49 - 0.0713i)T + (30.8 + 2.94i)T^{2}
37 1+(4.13−2.94i)T+(12.1−34.9i)T2 1 + (4.13 - 2.94i)T + (12.1 - 34.9i)T^{2}
41 1+(1.76−0.805i)T+(26.8−30.9i)T2 1 + (1.76 - 0.805i)T + (26.8 - 30.9i)T^{2}
43 1+(−2.52−3.92i)T+(−17.8+39.1i)T2 1 + (-2.52 - 3.92i)T + (-17.8 + 39.1i)T^{2}
47 1+(0.394+0.227i)T+(23.5+40.7i)T2 1 + (0.394 + 0.227i)T + (23.5 + 40.7i)T^{2}
53 1+(5.22−5.47i)T+(−2.52−52.9i)T2 1 + (5.22 - 5.47i)T + (-2.52 - 52.9i)T^{2}
59 1+(10.6−2.57i)T+(52.4−27.0i)T2 1 + (10.6 - 2.57i)T + (52.4 - 27.0i)T^{2}
61 1+(−11.1+5.73i)T+(35.3−49.6i)T2 1 + (-11.1 + 5.73i)T + (35.3 - 49.6i)T^{2}
67 1+(−0.820+4.25i)T+(−62.2−24.9i)T2 1 + (-0.820 + 4.25i)T + (-62.2 - 24.9i)T^{2}
71 1+(−7.18+8.29i)T+(−10.1−70.2i)T2 1 + (-7.18 + 8.29i)T + (-10.1 - 70.2i)T^{2}
73 1+(5.83+7.41i)T+(−17.2+70.9i)T2 1 + (5.83 + 7.41i)T + (-17.2 + 70.9i)T^{2}
79 1+(−5.95−6.24i)T+(−3.75+78.9i)T2 1 + (-5.95 - 6.24i)T + (-3.75 + 78.9i)T^{2}
83 1+(4.67−10.2i)T+(−54.3−62.7i)T2 1 + (4.67 - 10.2i)T + (-54.3 - 62.7i)T^{2}
89 1+(−0.0176−0.370i)T+(−88.5+8.45i)T2 1 + (-0.0176 - 0.370i)T + (-88.5 + 8.45i)T^{2}
97 1+(1.65+3.62i)T+(−63.5+73.3i)T2 1 + (1.65 + 3.62i)T + (-63.5 + 73.3i)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

−9.933760134188490575033356437534, −9.526734777723857199465231739370, −8.939619478367013937118808139014, −7.81979599820195764427126872073, −6.65032407547399233046143122426, −6.11427779745648018255379539497, −5.30795586691059175103876979911, −4.37789457198497824945870086153, −3.27198053089776028379344134041, −1.84997811399200594451732270481, 0.75453468003270610101362831473, 1.83895601451178212097388906614, 3.21483902365501257022292158762, 4.14742723328805101048355187053, 5.36356121880167865487740007978, 6.16259216846529563457484556962, 6.95197274608066849403859377240, 8.031371782639701225872118691000, 8.975514752636335754571452668374, 9.975843217940355175776243267785

Graph of the ZZ-function along the critical line