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Properties

Label 2-966-161.10-c1-0-10
Degree 22
Conductor 966966
Sign 0.353−0.935i0.353 - 0.935i
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 + (−0.000133 + 1.27e−5i)5-s + (0.989 − 0.142i)6-s + (1.26 + 2.32i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−3.16e−5 + 0.000130i)10-s + (−2.18 + 0.757i)11-s + (0.189 − 0.981i)12-s + (1.48 + 5.06i)13-s + (2.61 − 0.430i)14-s + (−7.26e−5 − 0.000113i)15-s + (0.235 + 0.971i)16-s + (−4.72 − 1.89i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−5.98e−5 + 5.71e−6i)5-s + (0.404 − 0.0580i)6-s + (0.476 + 0.879i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−1.00e−5 + 4.12e−5i)10-s + (−0.659 + 0.228i)11-s + (0.0546 − 0.283i)12-s + (0.412 + 1.40i)13-s + (0.697 − 0.114i)14-s + (−1.87e−5 − 2.91e−5i)15-s + (0.0589 + 0.242i)16-s + (−1.14 − 0.458i)17-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=((0.353−0.935i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=((0.353−0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: 0.353−0.935i0.353 - 0.935i
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), 0.353−0.935i)(2,\ 966,\ (\ :1/2),\ 0.353 - 0.935i)

Particular Values

L(1)L(1) ≈\approx 1.23228+0.851538i1.23228 + 0.851538i
L(12)L(\frac12) ≈\approx 1.23228+0.851538i1.23228 + 0.851538i
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+(−1.26−2.32i)T 1 + (-1.26 - 2.32i)T
23 1+(−3.59−3.17i)T 1 + (-3.59 - 3.17i)T
good5 1+(0.000133−1.27e−5i)T+(4.90−0.946i)T2 1 + (0.000133 - 1.27e-5i)T + (4.90 - 0.946i)T^{2}
11 1+(2.18−0.757i)T+(8.64−6.79i)T2 1 + (2.18 - 0.757i)T + (8.64 - 6.79i)T^{2}
13 1+(−1.48−5.06i)T+(−10.9+7.02i)T2 1 + (-1.48 - 5.06i)T + (-10.9 + 7.02i)T^{2}
17 1+(4.72+1.89i)T+(12.3+11.7i)T2 1 + (4.72 + 1.89i)T + (12.3 + 11.7i)T^{2}
19 1+(2.45−0.981i)T+(13.7−13.1i)T2 1 + (2.45 - 0.981i)T + (13.7 - 13.1i)T^{2}
29 1+(0.0478+0.332i)T+(−27.8+8.17i)T2 1 + (0.0478 + 0.332i)T + (-27.8 + 8.17i)T^{2}
31 1+(−0.479+0.0228i)T+(30.8−2.94i)T2 1 + (-0.479 + 0.0228i)T + (30.8 - 2.94i)T^{2}
37 1+(−7.64−5.44i)T+(12.1+34.9i)T2 1 + (-7.64 - 5.44i)T + (12.1 + 34.9i)T^{2}
41 1+(5.37+2.45i)T+(26.8+30.9i)T2 1 + (5.37 + 2.45i)T + (26.8 + 30.9i)T^{2}
43 1+(−1.37+2.14i)T+(−17.8−39.1i)T2 1 + (-1.37 + 2.14i)T + (-17.8 - 39.1i)T^{2}
47 1+(−1.41+0.817i)T+(23.5−40.7i)T2 1 + (-1.41 + 0.817i)T + (23.5 - 40.7i)T^{2}
53 1+(−7.60−7.97i)T+(−2.52+52.9i)T2 1 + (-7.60 - 7.97i)T + (-2.52 + 52.9i)T^{2}
59 1+(0.946+0.229i)T+(52.4+27.0i)T2 1 + (0.946 + 0.229i)T + (52.4 + 27.0i)T^{2}
61 1+(−5.85−3.01i)T+(35.3+49.6i)T2 1 + (-5.85 - 3.01i)T + (35.3 + 49.6i)T^{2}
67 1+(−0.312−1.62i)T+(−62.2+24.9i)T2 1 + (-0.312 - 1.62i)T + (-62.2 + 24.9i)T^{2}
71 1+(−2.20−2.54i)T+(−10.1+70.2i)T2 1 + (-2.20 - 2.54i)T + (-10.1 + 70.2i)T^{2}
73 1+(−8.53+10.8i)T+(−17.2−70.9i)T2 1 + (-8.53 + 10.8i)T + (-17.2 - 70.9i)T^{2}
79 1+(−1.82+1.90i)T+(−3.75−78.9i)T2 1 + (-1.82 + 1.90i)T + (-3.75 - 78.9i)T^{2}
83 1+(2.17+4.75i)T+(−54.3+62.7i)T2 1 + (2.17 + 4.75i)T + (-54.3 + 62.7i)T^{2}
89 1+(0.354−7.43i)T+(−88.5−8.45i)T2 1 + (0.354 - 7.43i)T + (-88.5 - 8.45i)T^{2}
97 1+(7.72−16.9i)T+(−63.5−73.3i)T2 1 + (7.72 - 16.9i)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

−10.20005614045243921523853866821, −9.262701361255510218846968868480, −8.871795200637956005481096747344, −7.909834612471419749976480092387, −6.67163144368746580113856188491, −5.59987376765114131060944016501, −4.72121072130280153922164416357, −3.98215523124711468455405989963, −2.65248536768127760166462667133, −1.86125002293713048936866202005, 0.60960658181908175762143311501, 2.37563692138627286770660352262, 3.63626353582881997852220951337, 4.61584908967294455396204888985, 5.63949366434759152310780688645, 6.53339332081747900768592466921, 7.35697517102502937576929113855, 8.189904031409061150353001319502, 8.527858897796249121780938411663, 9.844826822037738501795296542370

Graph of the ZZ-function along the critical line