Properties

Label 2-966-161.145-c1-0-12
Degree 22
Conductor 966966
Sign 0.1150.993i-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/2ΓC(s)L(s)=((0.1150.993i)Λ(2s)\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/2ΓC(s+1/2)L(s)=((0.1150.993i)Λ(1s)\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    =    237232 \cdot 3 \cdot 7 \cdot 23
Sign: 0.1150.993i-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), 0.1150.993i)(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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.3270.945i)T 1 + (-0.327 - 0.945i)T
3 1+(0.4580.888i)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+(4.160.397i)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+(1.515.17i)T+(10.97.02i)T2 1 + (1.51 - 5.17i)T + (-10.9 - 7.02i)T^{2}
17 1+(4.531.81i)T+(12.311.7i)T2 1 + (4.53 - 1.81i)T + (12.3 - 11.7i)T^{2}
19 1+(3.261.30i)T+(13.7+13.1i)T2 1 + (-3.26 - 1.30i)T + (13.7 + 13.1i)T^{2}
29 1+(0.7074.91i)T+(27.88.17i)T2 1 + (0.707 - 4.91i)T + (-27.8 - 8.17i)T^{2}
31 1+(7.820.372i)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+(12.134.9i)T2 1 + (-0.879 + 0.626i)T + (12.1 - 34.9i)T^{2}
41 1+(2.621.19i)T+(26.830.9i)T2 1 + (2.62 - 1.19i)T + (26.8 - 30.9i)T^{2}
43 1+(3.345.20i)T+(17.8+39.1i)T2 1 + (-3.34 - 5.20i)T + (-17.8 + 39.1i)T^{2}
47 1+(6.613.81i)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+(2.5252.9i)T2 1 + (-5.18 + 5.43i)T + (-2.52 - 52.9i)T^{2}
59 1+(9.752.36i)T+(52.427.0i)T2 1 + (9.75 - 2.36i)T + (52.4 - 27.0i)T^{2}
61 1+(5.152.65i)T+(35.349.6i)T2 1 + (5.15 - 2.65i)T + (35.3 - 49.6i)T^{2}
67 1+(2.09+10.8i)T+(62.224.9i)T2 1 + (-2.09 + 10.8i)T + (-62.2 - 24.9i)T^{2}
71 1+(0.01740.0201i)T+(10.170.2i)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+(54.362.7i)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+(0.1460.319i)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(1αj,pps)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