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Properties

Label 2-966-161.10-c1-0-28
Degree 22
Conductor 966966
Sign −0.977−0.210i-0.977 - 0.210i
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 + (−1.27 + 0.121i)5-s + (0.989 − 0.142i)6-s + (−0.678 + 2.55i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.302 + 1.24i)10-s + (−1.09 + 0.380i)11-s + (0.189 − 0.981i)12-s + (−1.87 − 6.37i)13-s + (2.19 + 1.47i)14-s + (−0.693 − 1.07i)15-s + (0.235 + 0.971i)16-s + (0.0653 + 0.0261i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.571 + 0.0545i)5-s + (0.404 − 0.0580i)6-s + (−0.256 + 0.966i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0956 + 0.394i)10-s + (−0.331 + 0.114i)11-s + (0.0546 − 0.283i)12-s + (−0.519 − 1.76i)13-s + (0.586 + 0.394i)14-s + (−0.179 − 0.278i)15-s + (0.0589 + 0.242i)16-s + (0.0158 + 0.00634i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((−0.977−0.210i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((−0.977−0.210i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: −0.977−0.210i-0.977 - 0.210i
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), −0.977−0.210i)(2,\ 966,\ (\ :1/2),\ -0.977 - 0.210i)

Particular Values

L(1)L(1) ≈\approx 0.00715958+0.0673010i0.00715958 + 0.0673010i
L(12)L(\frac12) ≈\approx 0.00715958+0.0673010i0.00715958 + 0.0673010i
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+(0.678−2.55i)T 1 + (0.678 - 2.55i)T
23 1+(0.864+4.71i)T 1 + (0.864 + 4.71i)T
good5 1+(1.27−0.121i)T+(4.90−0.946i)T2 1 + (1.27 - 0.121i)T + (4.90 - 0.946i)T^{2}
11 1+(1.09−0.380i)T+(8.64−6.79i)T2 1 + (1.09 - 0.380i)T + (8.64 - 6.79i)T^{2}
13 1+(1.87+6.37i)T+(−10.9+7.02i)T2 1 + (1.87 + 6.37i)T + (-10.9 + 7.02i)T^{2}
17 1+(−0.0653−0.0261i)T+(12.3+11.7i)T2 1 + (-0.0653 - 0.0261i)T + (12.3 + 11.7i)T^{2}
19 1+(3.05−1.22i)T+(13.7−13.1i)T2 1 + (3.05 - 1.22i)T + (13.7 - 13.1i)T^{2}
29 1+(−0.347−2.41i)T+(−27.8+8.17i)T2 1 + (-0.347 - 2.41i)T + (-27.8 + 8.17i)T^{2}
31 1+(2.40−0.114i)T+(30.8−2.94i)T2 1 + (2.40 - 0.114i)T + (30.8 - 2.94i)T^{2}
37 1+(9.68+6.89i)T+(12.1+34.9i)T2 1 + (9.68 + 6.89i)T + (12.1 + 34.9i)T^{2}
41 1+(6.49+2.96i)T+(26.8+30.9i)T2 1 + (6.49 + 2.96i)T + (26.8 + 30.9i)T^{2}
43 1+(4.18−6.51i)T+(−17.8−39.1i)T2 1 + (4.18 - 6.51i)T + (-17.8 - 39.1i)T^{2}
47 1+(−1.08+0.628i)T+(23.5−40.7i)T2 1 + (-1.08 + 0.628i)T + (23.5 - 40.7i)T^{2}
53 1+(−1.70−1.78i)T+(−2.52+52.9i)T2 1 + (-1.70 - 1.78i)T + (-2.52 + 52.9i)T^{2}
59 1+(−14.5−3.53i)T+(52.4+27.0i)T2 1 + (-14.5 - 3.53i)T + (52.4 + 27.0i)T^{2}
61 1+(−7.98−4.11i)T+(35.3+49.6i)T2 1 + (-7.98 - 4.11i)T + (35.3 + 49.6i)T^{2}
67 1+(−0.156−0.813i)T+(−62.2+24.9i)T2 1 + (-0.156 - 0.813i)T + (-62.2 + 24.9i)T^{2}
71 1+(−3.06−3.53i)T+(−10.1+70.2i)T2 1 + (-3.06 - 3.53i)T + (-10.1 + 70.2i)T^{2}
73 1+(1.16−1.47i)T+(−17.2−70.9i)T2 1 + (1.16 - 1.47i)T + (-17.2 - 70.9i)T^{2}
79 1+(11.8−12.4i)T+(−3.75−78.9i)T2 1 + (11.8 - 12.4i)T + (-3.75 - 78.9i)T^{2}
83 1+(2.25+4.92i)T+(−54.3+62.7i)T2 1 + (2.25 + 4.92i)T + (-54.3 + 62.7i)T^{2}
89 1+(0.0883−1.85i)T+(−88.5−8.45i)T2 1 + (0.0883 - 1.85i)T + (-88.5 - 8.45i)T^{2}
97 1+(−3.67+8.03i)T+(−63.5−73.3i)T2 1 + (-3.67 + 8.03i)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

−9.882144053124298745685471412438, −8.626229960824212770538129688929, −8.313729964101304964286884774106, −7.11465387110313013014048634651, −5.69978140787089497393285078815, −5.18489095873057399587672892433, −3.98084659331970810923433193374, −3.10094259394875831414749043509, −2.23592426715665195602102039452, −0.02624802339623618242754247880, 1.89858018621979962780088855811, 3.51668146596051145793951016624, 4.21341454636615972198223855727, 5.24258295293047397360889899170, 6.66394421563571644093063779422, 6.92045172593026062797342416800, 7.82529500311166714075418127691, 8.551445835089655471938318340410, 9.490799920443516579335746461377, 10.31967168154167157812281217104

Graph of the ZZ-function along the critical line