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Properties

Label 2-966-161.145-c1-0-12
Degree 22
Conductor 966966
Sign −0.115−0.993i-0.115 - 0.993i
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 + (4.16 + 0.397i)5-s + (−0.989 − 0.142i)6-s + (0.0805 − 2.64i)7-s + (−0.841 − 0.540i)8-s + (−0.580 − 0.814i)9-s + (0.986 + 4.06i)10-s + (−0.191 − 0.0663i)11-s + (−0.189 − 0.981i)12-s + (−1.51 + 5.17i)13-s + (2.52 − 0.788i)14-s + (−2.26 + 3.52i)15-s + (0.235 − 0.971i)16-s + (−4.53 + 1.81i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.668i)2-s + (−0.264 + 0.513i)3-s + (−0.393 + 0.309i)4-s + (1.86 + 0.177i)5-s + (−0.404 − 0.0580i)6-s + (0.0304 − 0.999i)7-s + (−0.297 − 0.191i)8-s + (−0.193 − 0.271i)9-s + (0.312 + 1.28i)10-s + (−0.0577 − 0.0199i)11-s + (−0.0546 − 0.283i)12-s + (−0.421 + 1.43i)13-s + (0.674 − 0.210i)14-s + (−0.584 + 0.909i)15-s + (0.0589 − 0.242i)16-s + (−1.09 + 0.439i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((−0.115−0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((−0.115−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: −0.115−0.993i-0.115 - 0.993i
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), −0.115−0.993i)(2,\ 966,\ (\ :1/2),\ -0.115 - 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.40353+1.57614i1.40353 + 1.57614i
L(12)L(\frac12) ≈\approx 1.40353+1.57614i1.40353 + 1.57614i
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+(−0.0805+2.64i)T 1 + (-0.0805 + 2.64i)T
23 1+(−4.79+0.0590i)T 1 + (-4.79 + 0.0590i)T
good5 1+(−4.16−0.397i)T+(4.90+0.946i)T2 1 + (-4.16 - 0.397i)T + (4.90 + 0.946i)T^{2}
11 1+(0.191+0.0663i)T+(8.64+6.79i)T2 1 + (0.191 + 0.0663i)T + (8.64 + 6.79i)T^{2}
13 1+(1.51−5.17i)T+(−10.9−7.02i)T2 1 + (1.51 - 5.17i)T + (-10.9 - 7.02i)T^{2}
17 1+(4.53−1.81i)T+(12.3−11.7i)T2 1 + (4.53 - 1.81i)T + (12.3 - 11.7i)T^{2}
19 1+(−3.26−1.30i)T+(13.7+13.1i)T2 1 + (-3.26 - 1.30i)T + (13.7 + 13.1i)T^{2}
29 1+(0.707−4.91i)T+(−27.8−8.17i)T2 1 + (0.707 - 4.91i)T + (-27.8 - 8.17i)T^{2}
31 1+(−7.82−0.372i)T+(30.8+2.94i)T2 1 + (-7.82 - 0.372i)T + (30.8 + 2.94i)T^{2}
37 1+(−0.879+0.626i)T+(12.1−34.9i)T2 1 + (-0.879 + 0.626i)T + (12.1 - 34.9i)T^{2}
41 1+(2.62−1.19i)T+(26.8−30.9i)T2 1 + (2.62 - 1.19i)T + (26.8 - 30.9i)T^{2}
43 1+(−3.34−5.20i)T+(−17.8+39.1i)T2 1 + (-3.34 - 5.20i)T + (-17.8 + 39.1i)T^{2}
47 1+(−6.61−3.81i)T+(23.5+40.7i)T2 1 + (-6.61 - 3.81i)T + (23.5 + 40.7i)T^{2}
53 1+(−5.18+5.43i)T+(−2.52−52.9i)T2 1 + (-5.18 + 5.43i)T + (-2.52 - 52.9i)T^{2}
59 1+(9.75−2.36i)T+(52.4−27.0i)T2 1 + (9.75 - 2.36i)T + (52.4 - 27.0i)T^{2}
61 1+(5.15−2.65i)T+(35.3−49.6i)T2 1 + (5.15 - 2.65i)T + (35.3 - 49.6i)T^{2}
67 1+(−2.09+10.8i)T+(−62.2−24.9i)T2 1 + (-2.09 + 10.8i)T + (-62.2 - 24.9i)T^{2}
71 1+(0.0174−0.0201i)T+(−10.1−70.2i)T2 1 + (0.0174 - 0.0201i)T + (-10.1 - 70.2i)T^{2}
73 1+(6.57+8.35i)T+(−17.2+70.9i)T2 1 + (6.57 + 8.35i)T + (-17.2 + 70.9i)T^{2}
79 1+(11.2+11.7i)T+(−3.75+78.9i)T2 1 + (11.2 + 11.7i)T + (-3.75 + 78.9i)T^{2}
83 1+(−2.10+4.61i)T+(−54.3−62.7i)T2 1 + (-2.10 + 4.61i)T + (-54.3 - 62.7i)T^{2}
89 1+(0.851+17.8i)T+(−88.5+8.45i)T2 1 + (0.851 + 17.8i)T + (-88.5 + 8.45i)T^{2}
97 1+(−0.146−0.319i)T+(−63.5+73.3i)T2 1 + (-0.146 - 0.319i)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

−10.20002275987643518175222047544, −9.296989870655800868987581621995, −8.943079773817086416515425759613, −7.41014134055542205028763651078, −6.60302939096078644575704072954, −6.11737633422254186102237775296, −4.96938010183692141537760458437, −4.41790060266786597967338410328, −2.97058208040942605884474541146, −1.54521121956127400412314571178, 1.04152375191357281607644659567, 2.44998082761253187135990805081, 2.70307141083735519411170467899, 4.83282920147797419995459804031, 5.48374741153471837107504663861, 6.02610783133198924828183897899, 7.05485132944321026551131966206, 8.427536209016755234714463516721, 9.157310918321573950048726752941, 9.850047583490697204394387924670

Graph of the ZZ-function along the critical line