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Properties

Label 2-966-161.10-c1-0-12
Degree 22
Conductor 966966
Sign −0.579−0.814i-0.579 - 0.814i
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 + (0.301 − 0.0288i)5-s + (−0.989 + 0.142i)6-s + (−0.249 + 2.63i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.0714 + 0.294i)10-s + (3.73 − 1.29i)11-s + (0.189 − 0.981i)12-s + (0.539 + 1.83i)13-s + (−2.40 − 1.09i)14-s + (0.163 + 0.254i)15-s + (0.235 + 0.971i)16-s + (3.12 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (0.134 − 0.0128i)5-s + (−0.404 + 0.0580i)6-s + (−0.0944 + 0.995i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0225 + 0.0931i)10-s + (1.12 − 0.389i)11-s + (0.0546 − 0.283i)12-s + (0.149 + 0.509i)13-s + (−0.643 − 0.293i)14-s + (0.0423 + 0.0658i)15-s + (0.0589 + 0.242i)16-s + (0.757 + 0.303i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((−0.579−0.814i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((−0.579−0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: −0.579−0.814i-0.579 - 0.814i
Analytic conductor: 7.713547.71354
Root analytic conductor: 2.777322.77732
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ966(493,⋅)\chi_{966} (493, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 966, ( :1/2), −0.579−0.814i)(2,\ 966,\ (\ :1/2),\ -0.579 - 0.814i)

Particular Values

L(1)L(1) ≈\approx 0.709996+1.37634i0.709996 + 1.37634i
L(12)L(\frac12) ≈\approx 0.709996+1.37634i0.709996 + 1.37634i
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+(0.249−2.63i)T 1 + (0.249 - 2.63i)T
23 1+(3.17−3.59i)T 1 + (3.17 - 3.59i)T
good5 1+(−0.301+0.0288i)T+(4.90−0.946i)T2 1 + (-0.301 + 0.0288i)T + (4.90 - 0.946i)T^{2}
11 1+(−3.73+1.29i)T+(8.64−6.79i)T2 1 + (-3.73 + 1.29i)T + (8.64 - 6.79i)T^{2}
13 1+(−0.539−1.83i)T+(−10.9+7.02i)T2 1 + (-0.539 - 1.83i)T + (-10.9 + 7.02i)T^{2}
17 1+(−3.12−1.25i)T+(12.3+11.7i)T2 1 + (-3.12 - 1.25i)T + (12.3 + 11.7i)T^{2}
19 1+(−7.95+3.18i)T+(13.7−13.1i)T2 1 + (-7.95 + 3.18i)T + (13.7 - 13.1i)T^{2}
29 1+(−0.182−1.26i)T+(−27.8+8.17i)T2 1 + (-0.182 - 1.26i)T + (-27.8 + 8.17i)T^{2}
31 1+(9.93−0.473i)T+(30.8−2.94i)T2 1 + (9.93 - 0.473i)T + (30.8 - 2.94i)T^{2}
37 1+(−3.33−2.37i)T+(12.1+34.9i)T2 1 + (-3.33 - 2.37i)T + (12.1 + 34.9i)T^{2}
41 1+(−2.25−1.03i)T+(26.8+30.9i)T2 1 + (-2.25 - 1.03i)T + (26.8 + 30.9i)T^{2}
43 1+(−2.60+4.05i)T+(−17.8−39.1i)T2 1 + (-2.60 + 4.05i)T + (-17.8 - 39.1i)T^{2}
47 1+(9.73−5.61i)T+(23.5−40.7i)T2 1 + (9.73 - 5.61i)T + (23.5 - 40.7i)T^{2}
53 1+(−7.27−7.62i)T+(−2.52+52.9i)T2 1 + (-7.27 - 7.62i)T + (-2.52 + 52.9i)T^{2}
59 1+(−7.18−1.74i)T+(52.4+27.0i)T2 1 + (-7.18 - 1.74i)T + (52.4 + 27.0i)T^{2}
61 1+(5.36+2.76i)T+(35.3+49.6i)T2 1 + (5.36 + 2.76i)T + (35.3 + 49.6i)T^{2}
67 1+(−0.294−1.53i)T+(−62.2+24.9i)T2 1 + (-0.294 - 1.53i)T + (-62.2 + 24.9i)T^{2}
71 1+(−4.04−4.67i)T+(−10.1+70.2i)T2 1 + (-4.04 - 4.67i)T + (-10.1 + 70.2i)T^{2}
73 1+(−0.581+0.739i)T+(−17.2−70.9i)T2 1 + (-0.581 + 0.739i)T + (-17.2 - 70.9i)T^{2}
79 1+(6.89−7.23i)T+(−3.75−78.9i)T2 1 + (6.89 - 7.23i)T + (-3.75 - 78.9i)T^{2}
83 1+(−4.53−9.93i)T+(−54.3+62.7i)T2 1 + (-4.53 - 9.93i)T + (-54.3 + 62.7i)T^{2}
89 1+(0.299−6.29i)T+(−88.5−8.45i)T2 1 + (0.299 - 6.29i)T + (-88.5 - 8.45i)T^{2}
97 1+(−7.30+15.9i)T+(−63.5−73.3i)T2 1 + (-7.30 + 15.9i)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

−9.750586023507702009595575827329, −9.470936065035187957018659342451, −8.796083870956303661085256889625, −7.86149868675810298551745556798, −6.97442477061786258722338477487, −5.80563788706914994466969167726, −5.44417063103865652265726163675, −4.08774921152955187812102881238, −3.16575051864059297332628764599, −1.56029148559038653978506161673, 0.839727631655079285058932044915, 1.89213651213315427646016718696, 3.38837229891905904620775558061, 3.94931956271532885461736624197, 5.34748829190727193432093552678, 6.42652508349402057994619223084, 7.52768030800214936964048111393, 7.85216457287500443831935697977, 9.131827706063689074479124536615, 9.788116220494236881735461934812

Graph of the ZZ-function along the critical line