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Properties

Label 2-666-9.4-c1-0-13
Degree 22
Conductor 666666
Sign 0.986+0.163i0.986 + 0.163i
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  + (−0.5 − 0.866i)2-s + (1.41 − 1.00i)3-s + (−0.499 + 0.866i)4-s + (−1.74 + 3.02i)5-s + (−1.57 − 0.723i)6-s + (−1.43 − 2.47i)7-s + 0.999·8-s + (0.995 − 2.82i)9-s + 3.49·10-s + (2.71 + 4.70i)11-s + (0.160 + 1.72i)12-s + (−1.81 + 3.14i)13-s + (−1.43 + 2.47i)14-s + (0.559 + 6.02i)15-s + (−0.5 − 0.866i)16-s + 4.84·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.816 − 0.577i)3-s + (−0.249 + 0.433i)4-s + (−0.780 + 1.35i)5-s + (−0.642 − 0.295i)6-s + (−0.540 − 0.936i)7-s + 0.353·8-s + (0.331 − 0.943i)9-s + 1.10·10-s + (0.819 + 1.41i)11-s + (0.0462 + 0.497i)12-s + (−0.503 + 0.871i)13-s + (−0.382 + 0.662i)14-s + (0.144 + 1.55i)15-s + (−0.125 − 0.216i)16-s + 1.17·17-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.986+0.163i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.986+0.163i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.986+0.163i0.986 + 0.163i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(445,⋅)\chi_{666} (445, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.986+0.163i)(2,\ 666,\ (\ :1/2),\ 0.986 + 0.163i)

Particular Values

L(1)L(1) ≈\approx 1.39713−0.114667i1.39713 - 0.114667i
L(12)L(\frac12) ≈\approx 1.39713−0.114667i1.39713 - 0.114667i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(−1.41+1.00i)T 1 + (-1.41 + 1.00i)T
37 1+T 1 + T
good5 1+(1.74−3.02i)T+(−2.5−4.33i)T2 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2}
7 1+(1.43+2.47i)T+(−3.5+6.06i)T2 1 + (1.43 + 2.47i)T + (-3.5 + 6.06i)T^{2}
11 1+(−2.71−4.70i)T+(−5.5+9.52i)T2 1 + (-2.71 - 4.70i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.81−3.14i)T+(−6.5−11.2i)T2 1 + (1.81 - 3.14i)T + (-6.5 - 11.2i)T^{2}
17 1−4.84T+17T2 1 - 4.84T + 17T^{2}
19 1−7.31T+19T2 1 - 7.31T + 19T^{2}
23 1+(−3.34+5.79i)T+(−11.5−19.9i)T2 1 + (-3.34 + 5.79i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.90−5.03i)T+(−14.5+25.1i)T2 1 + (-2.90 - 5.03i)T + (-14.5 + 25.1i)T^{2}
31 1+(4.48−7.76i)T+(−15.5−26.8i)T2 1 + (4.48 - 7.76i)T + (-15.5 - 26.8i)T^{2}
41 1+(−0.330+0.571i)T+(−20.5−35.5i)T2 1 + (-0.330 + 0.571i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.88−4.99i)T+(−21.5+37.2i)T2 1 + (-2.88 - 4.99i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3.70−6.41i)T+(−23.5+40.7i)T2 1 + (-3.70 - 6.41i)T + (-23.5 + 40.7i)T^{2}
53 1−3.88T+53T2 1 - 3.88T + 53T^{2}
59 1+(0.782−1.35i)T+(−29.5−51.0i)T2 1 + (0.782 - 1.35i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.24+7.35i)T+(−30.5+52.8i)T2 1 + (4.24 + 7.35i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.185−0.321i)T+(−33.5−58.0i)T2 1 + (0.185 - 0.321i)T + (-33.5 - 58.0i)T^{2}
71 1+0.584T+71T2 1 + 0.584T + 71T^{2}
73 1+11.9T+73T2 1 + 11.9T + 73T^{2}
79 1+(−2.06−3.58i)T+(−39.5+68.4i)T2 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.42+2.47i)T+(−41.5+71.8i)T2 1 + (1.42 + 2.47i)T + (-41.5 + 71.8i)T^{2}
89 1+3.31T+89T2 1 + 3.31T + 89T^{2}
97 1+(6.41+11.1i)T+(−48.5+84.0i)T2 1 + (6.41 + 11.1i)T + (-48.5 + 84.0i)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

−10.31058625985739434785911515318, −9.771517131721641902239574279034, −8.924394605085166162959759845176, −7.51130822825901969318669177747, −7.22206923908932397745308149480, −6.71280922001146503796294053942, −4.46872965591701100703466060216, −3.49096901445870665068090421653, −2.87625005222721337765939434940, −1.34883856785028758844248695682, 0.932013565805815905743477714896, 3.07792127890063403872553009489, 3.92579193300912308887585462947, 5.39786468528094352262762924403, 5.61359858826853520372939162946, 7.53289193637281685031272502078, 7.983535597643077156388644626594, 9.006763274266662422581981555070, 9.189936829334496374921773100558, 10.10512761828052494880903234910

Graph of the ZZ-function along the critical line