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Properties

Label 2-966-161.145-c1-0-4
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(145,⋅)\chi_{966} (145, \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

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

Graph of the ZZ-function along the critical line