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Properties

Label 2-966-161.10-c1-0-14
Degree 22
Conductor 966966
Sign 0.629+0.776i0.629 + 0.776i
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 + (−1.97 + 0.188i)5-s + (−0.989 + 0.142i)6-s + (−0.675 + 2.55i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.467 + 1.92i)10-s + (2.38 − 0.826i)11-s + (−0.189 + 0.981i)12-s + (0.570 + 1.94i)13-s + (2.19 + 1.47i)14-s + (1.07 + 1.66i)15-s + (0.235 + 0.971i)16-s + (3.52 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.882 + 0.0843i)5-s + (−0.404 + 0.0580i)6-s + (−0.255 + 0.966i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.147 + 0.609i)10-s + (0.719 − 0.249i)11-s + (−0.0546 + 0.283i)12-s + (0.158 + 0.538i)13-s + (0.587 + 0.394i)14-s + (0.276 + 0.430i)15-s + (0.0589 + 0.242i)16-s + (0.856 + 0.342i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((0.629+0.776i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((0.629+0.776i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: 0.629+0.776i0.629 + 0.776i
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), 0.629+0.776i)(2,\ 966,\ (\ :1/2),\ 0.629 + 0.776i)

Particular Values

L(1)L(1) ≈\approx 1.19029−0.567362i1.19029 - 0.567362i
L(12)L(\frac12) ≈\approx 1.19029−0.567362i1.19029 - 0.567362i
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+(0.675−2.55i)T 1 + (0.675 - 2.55i)T
23 1+(−2.28+4.21i)T 1 + (-2.28 + 4.21i)T
good5 1+(1.97−0.188i)T+(4.90−0.946i)T2 1 + (1.97 - 0.188i)T + (4.90 - 0.946i)T^{2}
11 1+(−2.38+0.826i)T+(8.64−6.79i)T2 1 + (-2.38 + 0.826i)T + (8.64 - 6.79i)T^{2}
13 1+(−0.570−1.94i)T+(−10.9+7.02i)T2 1 + (-0.570 - 1.94i)T + (-10.9 + 7.02i)T^{2}
17 1+(−3.52−1.41i)T+(12.3+11.7i)T2 1 + (-3.52 - 1.41i)T + (12.3 + 11.7i)T^{2}
19 1+(−4.92+1.97i)T+(13.7−13.1i)T2 1 + (-4.92 + 1.97i)T + (13.7 - 13.1i)T^{2}
29 1+(0.905+6.29i)T+(−27.8+8.17i)T2 1 + (0.905 + 6.29i)T + (-27.8 + 8.17i)T^{2}
31 1+(−0.745+0.0355i)T+(30.8−2.94i)T2 1 + (-0.745 + 0.0355i)T + (30.8 - 2.94i)T^{2}
37 1+(0.924+0.658i)T+(12.1+34.9i)T2 1 + (0.924 + 0.658i)T + (12.1 + 34.9i)T^{2}
41 1+(−10.0−4.58i)T+(26.8+30.9i)T2 1 + (-10.0 - 4.58i)T + (26.8 + 30.9i)T^{2}
43 1+(2.01−3.12i)T+(−17.8−39.1i)T2 1 + (2.01 - 3.12i)T + (-17.8 - 39.1i)T^{2}
47 1+(1.95−1.12i)T+(23.5−40.7i)T2 1 + (1.95 - 1.12i)T + (23.5 - 40.7i)T^{2}
53 1+(−6.02−6.31i)T+(−2.52+52.9i)T2 1 + (-6.02 - 6.31i)T + (-2.52 + 52.9i)T^{2}
59 1+(−8.15−1.97i)T+(52.4+27.0i)T2 1 + (-8.15 - 1.97i)T + (52.4 + 27.0i)T^{2}
61 1+(−7.45−3.84i)T+(35.3+49.6i)T2 1 + (-7.45 - 3.84i)T + (35.3 + 49.6i)T^{2}
67 1+(−1.65−8.59i)T+(−62.2+24.9i)T2 1 + (-1.65 - 8.59i)T + (-62.2 + 24.9i)T^{2}
71 1+(5.71+6.59i)T+(−10.1+70.2i)T2 1 + (5.71 + 6.59i)T + (-10.1 + 70.2i)T^{2}
73 1+(−3.00+3.82i)T+(−17.2−70.9i)T2 1 + (-3.00 + 3.82i)T + (-17.2 - 70.9i)T^{2}
79 1+(−6.24+6.55i)T+(−3.75−78.9i)T2 1 + (-6.24 + 6.55i)T + (-3.75 - 78.9i)T^{2}
83 1+(−7.56−16.5i)T+(−54.3+62.7i)T2 1 + (-7.56 - 16.5i)T + (-54.3 + 62.7i)T^{2}
89 1+(−0.832+17.4i)T+(−88.5−8.45i)T2 1 + (-0.832 + 17.4i)T + (-88.5 - 8.45i)T^{2}
97 1+(0.578−1.26i)T+(−63.5−73.3i)T2 1 + (0.578 - 1.26i)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

−9.889334539354473885170135822536, −9.131886692404763804322357639984, −8.300587410512099475955310083312, −7.41293818721344955253780011945, −6.34262043134195637168091905958, −5.60429413527479801995137751401, −4.44209608957512951630157923140, −3.45306353966649917004471131895, −2.43308518239592368868532918198, −0.954186430479940530279806637702, 0.869865639726113447999708623039, 3.49739171646864283746447508219, 3.73995773924846783384398651816, 4.93934814535067744137021991821, 5.69928475845703962917757290248, 6.94629749769757071770165024993, 7.45566978361457130425923938505, 8.272754098308042510963291920754, 9.388423394930079841426306643720, 9.983956036869691893853387922376

Graph of the ZZ-function along the critical line