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Properties

Label 2-966-161.10-c1-0-26
Degree 22
Conductor 966966
Sign −0.230+0.973i-0.230 + 0.973i
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 + (−1.17 + 0.111i)5-s + (0.989 − 0.142i)6-s + (1.80 − 1.93i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.277 − 1.14i)10-s + (0.654 − 0.226i)11-s + (−0.189 + 0.981i)12-s + (−0.727 − 2.47i)13-s + (1.23 + 2.34i)14-s + (0.635 + 0.989i)15-s + (0.235 + 0.971i)16-s + (3.55 + 1.42i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.523 + 0.0500i)5-s + (0.404 − 0.0580i)6-s + (0.683 − 0.729i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.0876 − 0.361i)10-s + (0.197 − 0.0682i)11-s + (−0.0546 + 0.283i)12-s + (−0.201 − 0.687i)13-s + (0.329 + 0.625i)14-s + (0.164 + 0.255i)15-s + (0.0589 + 0.242i)16-s + (0.863 + 0.345i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((−0.230+0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((−0.230+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: −0.230+0.973i-0.230 + 0.973i
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), −0.230+0.973i)(2,\ 966,\ (\ :1/2),\ -0.230 + 0.973i)

Particular Values

L(1)L(1) ≈\approx 0.434558−0.549313i0.434558 - 0.549313i
L(12)L(\frac12) ≈\approx 0.434558−0.549313i0.434558 - 0.549313i
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+(−1.80+1.93i)T 1 + (-1.80 + 1.93i)T
23 1+(−3.80+2.91i)T 1 + (-3.80 + 2.91i)T
good5 1+(1.17−0.111i)T+(4.90−0.946i)T2 1 + (1.17 - 0.111i)T + (4.90 - 0.946i)T^{2}
11 1+(−0.654+0.226i)T+(8.64−6.79i)T2 1 + (-0.654 + 0.226i)T + (8.64 - 6.79i)T^{2}
13 1+(0.727+2.47i)T+(−10.9+7.02i)T2 1 + (0.727 + 2.47i)T + (-10.9 + 7.02i)T^{2}
17 1+(−3.55−1.42i)T+(12.3+11.7i)T2 1 + (-3.55 - 1.42i)T + (12.3 + 11.7i)T^{2}
19 1+(6.95−2.78i)T+(13.7−13.1i)T2 1 + (6.95 - 2.78i)T + (13.7 - 13.1i)T^{2}
29 1+(1.02+7.14i)T+(−27.8+8.17i)T2 1 + (1.02 + 7.14i)T + (-27.8 + 8.17i)T^{2}
31 1+(6.04−0.288i)T+(30.8−2.94i)T2 1 + (6.04 - 0.288i)T + (30.8 - 2.94i)T^{2}
37 1+(5.16+3.67i)T+(12.1+34.9i)T2 1 + (5.16 + 3.67i)T + (12.1 + 34.9i)T^{2}
41 1+(−7.63−3.48i)T+(26.8+30.9i)T2 1 + (-7.63 - 3.48i)T + (26.8 + 30.9i)T^{2}
43 1+(−4.24+6.60i)T+(−17.8−39.1i)T2 1 + (-4.24 + 6.60i)T + (-17.8 - 39.1i)T^{2}
47 1+(5.53−3.19i)T+(23.5−40.7i)T2 1 + (5.53 - 3.19i)T + (23.5 - 40.7i)T^{2}
53 1+(7.46+7.83i)T+(−2.52+52.9i)T2 1 + (7.46 + 7.83i)T + (-2.52 + 52.9i)T^{2}
59 1+(2.11+0.513i)T+(52.4+27.0i)T2 1 + (2.11 + 0.513i)T + (52.4 + 27.0i)T^{2}
61 1+(0.602+0.310i)T+(35.3+49.6i)T2 1 + (0.602 + 0.310i)T + (35.3 + 49.6i)T^{2}
67 1+(2.64+13.7i)T+(−62.2+24.9i)T2 1 + (2.64 + 13.7i)T + (-62.2 + 24.9i)T^{2}
71 1+(3.89+4.49i)T+(−10.1+70.2i)T2 1 + (3.89 + 4.49i)T + (-10.1 + 70.2i)T^{2}
73 1+(−5.29+6.73i)T+(−17.2−70.9i)T2 1 + (-5.29 + 6.73i)T + (-17.2 - 70.9i)T^{2}
79 1+(11.2−11.8i)T+(−3.75−78.9i)T2 1 + (11.2 - 11.8i)T + (-3.75 - 78.9i)T^{2}
83 1+(−2.04−4.48i)T+(−54.3+62.7i)T2 1 + (-2.04 - 4.48i)T + (-54.3 + 62.7i)T^{2}
89 1+(−0.388+8.15i)T+(−88.5−8.45i)T2 1 + (-0.388 + 8.15i)T + (-88.5 - 8.45i)T^{2}
97 1+(−1.11+2.44i)T+(−63.5−73.3i)T2 1 + (-1.11 + 2.44i)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

−9.784965505166258890201765926160, −8.597956862644977783958215508318, −7.87261596973841474229228924080, −7.48106861407341316870247296717, −6.43012302502685174183374555016, −5.63127178056954609036185876790, −4.55465750839058266031212805030, −3.64666138420558248593580485960, −1.84046342753762488601621112057, −0.37131204319453143456454463615, 1.59998335692542206630878742481, 2.88846332165583835966568735474, 4.04299523379254643996524721649, 4.81084095588464009052192723594, 5.71387017628060267668929319763, 7.00646392350189213602505186206, 7.940131919893297476703008075650, 8.947529208428155416150292570521, 9.253105272574463834463437093492, 10.41821828892600454370044791883

Graph of the ZZ-function along the critical line