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Properties

Label 2-666-111.2-c1-0-12
Degree 22
Conductor 666666
Sign −0.982−0.187i-0.982 - 0.187i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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  + (0.0871 − 0.996i)2-s + (−0.984 − 0.173i)4-s + (0.0741 + 0.0345i)5-s + (−1.92 + 0.700i)7-s + (−0.258 + 0.965i)8-s + (0.0408 − 0.0708i)10-s + (−0.0467 − 0.0809i)11-s + (−3.00 − 4.29i)13-s + (0.530 + 1.97i)14-s + (0.939 + 0.342i)16-s + (2.94 − 4.21i)17-s + (−7.61 + 0.666i)19-s + (−0.0669 − 0.0468i)20-s + (−0.0847 + 0.0395i)22-s + (−6.72 + 1.80i)23-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (0.0331 + 0.0154i)5-s + (−0.727 + 0.264i)7-s + (−0.0915 + 0.341i)8-s + (0.0129 − 0.0223i)10-s + (−0.0140 − 0.0244i)11-s + (−0.833 − 1.19i)13-s + (0.141 + 0.529i)14-s + (0.234 + 0.0855i)16-s + (0.715 − 1.02i)17-s + (−1.74 + 0.152i)19-s + (−0.0149 − 0.0104i)20-s + (−0.0180 + 0.00842i)22-s + (−1.40 + 0.375i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((−0.982−0.187i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((−0.982−0.187i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −0.982−0.187i-0.982 - 0.187i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(557,⋅)\chi_{666} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), −0.982−0.187i)(2,\ 666,\ (\ :1/2),\ -0.982 - 0.187i)

Particular Values

L(1)L(1) ≈\approx 0.0443399+0.468175i0.0443399 + 0.468175i
L(12)L(\frac12) ≈\approx 0.0443399+0.468175i0.0443399 + 0.468175i
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+(−0.0871+0.996i)T 1 + (-0.0871 + 0.996i)T
3 1 1
37 1+(6.08+0.146i)T 1 + (6.08 + 0.146i)T
good5 1+(−0.0741−0.0345i)T+(3.21+3.83i)T2 1 + (-0.0741 - 0.0345i)T + (3.21 + 3.83i)T^{2}
7 1+(1.92−0.700i)T+(5.36−4.49i)T2 1 + (1.92 - 0.700i)T + (5.36 - 4.49i)T^{2}
11 1+(0.0467+0.0809i)T+(−5.5+9.52i)T2 1 + (0.0467 + 0.0809i)T + (-5.5 + 9.52i)T^{2}
13 1+(3.00+4.29i)T+(−4.44+12.2i)T2 1 + (3.00 + 4.29i)T + (-4.44 + 12.2i)T^{2}
17 1+(−2.94+4.21i)T+(−5.81−15.9i)T2 1 + (-2.94 + 4.21i)T + (-5.81 - 15.9i)T^{2}
19 1+(7.61−0.666i)T+(18.7−3.29i)T2 1 + (7.61 - 0.666i)T + (18.7 - 3.29i)T^{2}
23 1+(6.72−1.80i)T+(19.9−11.5i)T2 1 + (6.72 - 1.80i)T + (19.9 - 11.5i)T^{2}
29 1+(−3.03−0.813i)T+(25.1+14.5i)T2 1 + (-3.03 - 0.813i)T + (25.1 + 14.5i)T^{2}
31 1+(0.356+0.356i)T+31iT2 1 + (0.356 + 0.356i)T + 31iT^{2}
41 1+(0.847−4.80i)T+(−38.5−14.0i)T2 1 + (0.847 - 4.80i)T + (-38.5 - 14.0i)T^{2}
43 1+(−5.17+5.17i)T−43iT2 1 + (-5.17 + 5.17i)T - 43iT^{2}
47 1+(−7.78−4.49i)T+(23.5+40.7i)T2 1 + (-7.78 - 4.49i)T + (23.5 + 40.7i)T^{2}
53 1+(−2.03+5.57i)T+(−40.6−34.0i)T2 1 + (-2.03 + 5.57i)T + (-40.6 - 34.0i)T^{2}
59 1+(3.74+8.02i)T+(−37.9+45.1i)T2 1 + (3.74 + 8.02i)T + (-37.9 + 45.1i)T^{2}
61 1+(4.71−3.30i)T+(20.8−57.3i)T2 1 + (4.71 - 3.30i)T + (20.8 - 57.3i)T^{2}
67 1+(−1.89−5.19i)T+(−51.3+43.0i)T2 1 + (-1.89 - 5.19i)T + (-51.3 + 43.0i)T^{2}
71 1+(6.74−8.03i)T+(−12.3−69.9i)T2 1 + (6.74 - 8.03i)T + (-12.3 - 69.9i)T^{2}
73 1+4.72iT−73T2 1 + 4.72iT - 73T^{2}
79 1+(−5.96+12.7i)T+(−50.7−60.5i)T2 1 + (-5.96 + 12.7i)T + (-50.7 - 60.5i)T^{2}
83 1+(−11.2+1.97i)T+(77.9−28.3i)T2 1 + (-11.2 + 1.97i)T + (77.9 - 28.3i)T^{2}
89 1+(5.09−2.37i)T+(57.2−68.1i)T2 1 + (5.09 - 2.37i)T + (57.2 - 68.1i)T^{2}
97 1+(−0.309−1.15i)T+(−84.0+48.5i)T2 1 + (-0.309 - 1.15i)T + (-84.0 + 48.5i)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

−10.12892299800449118351099529290, −9.480164935579498637010334511778, −8.405296672283928636469841890573, −7.58481135663173512934574084062, −6.30714915368743356450811219461, −5.46167355843715384840458740520, −4.31549532993641149289436657835, −3.16488043863363606928252086946, −2.22707358254215702149217997133, −0.22964306331364436031938028571, 2.08351206386321628159675042440, 3.75329692724989981046601765795, 4.48004757456554392187967698463, 5.83867581768252846693972360491, 6.50932158918030155536158323569, 7.35671057682421965028574225570, 8.321395001504077670681811988523, 9.177956126990232658955507237817, 10.02365645956319350730931726252, 10.72209254345196358494252260814

Graph of the ZZ-function along the critical line