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Properties

Label 2-966-161.145-c1-0-14
Degree 22
Conductor 966966
Sign 0.146+0.989i0.146 + 0.989i
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 + (−1.63 − 0.156i)5-s + (−0.989 − 0.142i)6-s + (−0.689 + 2.55i)7-s + (0.841 + 0.540i)8-s + (−0.580 − 0.814i)9-s + (0.387 + 1.59i)10-s + (2.48 + 0.859i)11-s + (0.189 + 0.981i)12-s + (0.647 − 2.20i)13-s + (2.63 − 0.183i)14-s + (−0.888 + 1.38i)15-s + (0.235 − 0.971i)16-s + (1.98 − 0.795i)17-s + ⋯
L(s)  = 1  + (−0.231 − 0.668i)2-s + (0.264 − 0.513i)3-s + (−0.393 + 0.309i)4-s + (−0.731 − 0.0698i)5-s + (−0.404 − 0.0580i)6-s + (−0.260 + 0.965i)7-s + (0.297 + 0.191i)8-s + (−0.193 − 0.271i)9-s + (0.122 + 0.505i)10-s + (0.748 + 0.259i)11-s + (0.0546 + 0.283i)12-s + (0.179 − 0.611i)13-s + (0.705 − 0.0490i)14-s + (−0.229 + 0.356i)15-s + (0.0589 − 0.242i)16-s + (0.482 − 0.192i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((0.146+0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((0.146+0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: 0.146+0.989i0.146 + 0.989i
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), 0.146+0.989i)(2,\ 966,\ (\ :1/2),\ 0.146 + 0.989i)

Particular Values

L(1)L(1) ≈\approx 0.970431−0.837529i0.970431 - 0.837529i
L(12)L(\frac12) ≈\approx 0.970431−0.837529i0.970431 - 0.837529i
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+(0.689−2.55i)T 1 + (0.689 - 2.55i)T
23 1+(−4.61+1.31i)T 1 + (-4.61 + 1.31i)T
good5 1+(1.63+0.156i)T+(4.90+0.946i)T2 1 + (1.63 + 0.156i)T + (4.90 + 0.946i)T^{2}
11 1+(−2.48−0.859i)T+(8.64+6.79i)T2 1 + (-2.48 - 0.859i)T + (8.64 + 6.79i)T^{2}
13 1+(−0.647+2.20i)T+(−10.9−7.02i)T2 1 + (-0.647 + 2.20i)T + (-10.9 - 7.02i)T^{2}
17 1+(−1.98+0.795i)T+(12.3−11.7i)T2 1 + (-1.98 + 0.795i)T + (12.3 - 11.7i)T^{2}
19 1+(−1.53−0.616i)T+(13.7+13.1i)T2 1 + (-1.53 - 0.616i)T + (13.7 + 13.1i)T^{2}
29 1+(−1.10+7.65i)T+(−27.8−8.17i)T2 1 + (-1.10 + 7.65i)T + (-27.8 - 8.17i)T^{2}
31 1+(−5.68−0.270i)T+(30.8+2.94i)T2 1 + (-5.68 - 0.270i)T + (30.8 + 2.94i)T^{2}
37 1+(1.22−0.873i)T+(12.1−34.9i)T2 1 + (1.22 - 0.873i)T + (12.1 - 34.9i)T^{2}
41 1+(−7.20+3.28i)T+(26.8−30.9i)T2 1 + (-7.20 + 3.28i)T + (26.8 - 30.9i)T^{2}
43 1+(3.37+5.25i)T+(−17.8+39.1i)T2 1 + (3.37 + 5.25i)T + (-17.8 + 39.1i)T^{2}
47 1+(−9.53−5.50i)T+(23.5+40.7i)T2 1 + (-9.53 - 5.50i)T + (23.5 + 40.7i)T^{2}
53 1+(−4.68+4.91i)T+(−2.52−52.9i)T2 1 + (-4.68 + 4.91i)T + (-2.52 - 52.9i)T^{2}
59 1+(6.85−1.66i)T+(52.4−27.0i)T2 1 + (6.85 - 1.66i)T + (52.4 - 27.0i)T^{2}
61 1+(2.19−1.13i)T+(35.3−49.6i)T2 1 + (2.19 - 1.13i)T + (35.3 - 49.6i)T^{2}
67 1+(−0.173+0.902i)T+(−62.2−24.9i)T2 1 + (-0.173 + 0.902i)T + (-62.2 - 24.9i)T^{2}
71 1+(−8.05+9.29i)T+(−10.1−70.2i)T2 1 + (-8.05 + 9.29i)T + (-10.1 - 70.2i)T^{2}
73 1+(−2.94−3.73i)T+(−17.2+70.9i)T2 1 + (-2.94 - 3.73i)T + (-17.2 + 70.9i)T^{2}
79 1+(1.28+1.34i)T+(−3.75+78.9i)T2 1 + (1.28 + 1.34i)T + (-3.75 + 78.9i)T^{2}
83 1+(2.18−4.79i)T+(−54.3−62.7i)T2 1 + (2.18 - 4.79i)T + (-54.3 - 62.7i)T^{2}
89 1+(−0.521−10.9i)T+(−88.5+8.45i)T2 1 + (-0.521 - 10.9i)T + (-88.5 + 8.45i)T^{2}
97 1+(−3.10−6.78i)T+(−63.5+73.3i)T2 1 + (-3.10 - 6.78i)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

−9.690056893466037340663293392557, −9.044652082715694500450223845402, −8.217479086533986405307424240701, −7.58467775555632024249800439179, −6.47658272503214532726010216498, −5.48406840643628321847141555276, −4.23342431244343478878584270562, −3.24059060800164518629920918911, −2.31693116344996354129632290322, −0.822150829204916605794371922646, 1.09573763596627927241652883158, 3.24531757031976636951450995319, 4.01325136248227037329616399989, 4.81835540996464076663644545133, 6.06981345103668434002590714368, 7.01885407631696375842887670733, 7.59337901327504991928974839091, 8.567608112278383640436468739406, 9.263382550981303211640928232454, 10.05621254880222483584783619797

Graph of the ZZ-function along the critical line