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Properties

Label 2-418-11.5-c1-0-11
Degree 22
Conductor 418418
Sign 0.569−0.821i0.569 - 0.821i
Analytic cond. 3.337743.33774
Root an. cond. 1.826951.82695
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  + (0.309 + 0.951i)2-s + (1.61 + 1.17i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 1.53i)5-s + (−0.618 + 1.90i)6-s + (3.92 − 2.85i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 1.61·10-s + (−0.809 + 3.21i)11-s − 1.99·12-s + (0.763 + 2.35i)13-s + (3.92 + 2.85i)14-s + (2.61 − 1.90i)15-s + (0.309 − 0.951i)16-s + (1.04 − 3.21i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.934 + 0.678i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.688i)5-s + (−0.252 + 0.776i)6-s + (1.48 − 1.07i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + 0.511·10-s + (−0.243 + 0.969i)11-s − 0.577·12-s + (0.211 + 0.652i)13-s + (1.04 + 0.762i)14-s + (0.675 − 0.491i)15-s + (0.0772 − 0.237i)16-s + (0.253 − 0.780i)17-s + ⋯

Functional equation

Λ(s)=(418s/2 ΓC(s) L(s)=((0.569−0.821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(418s/2 ΓC(s+1/2) L(s)=((0.569−0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 418418    =    2⋅11⋅192 \cdot 11 \cdot 19
Sign: 0.569−0.821i0.569 - 0.821i
Analytic conductor: 3.337743.33774
Root analytic conductor: 1.826951.82695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ418(115,⋅)\chi_{418} (115, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 418, ( :1/2), 0.569−0.821i)(2,\ 418,\ (\ :1/2),\ 0.569 - 0.821i)

Particular Values

L(1)L(1) ≈\approx 1.97641+1.03455i1.97641 + 1.03455i
L(12)L(\frac12) ≈\approx 1.97641+1.03455i1.97641 + 1.03455i
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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
11 1+(0.809−3.21i)T 1 + (0.809 - 3.21i)T
19 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
good3 1+(−1.61−1.17i)T+(0.927+2.85i)T2 1 + (-1.61 - 1.17i)T + (0.927 + 2.85i)T^{2}
5 1+(−0.5+1.53i)T+(−4.04−2.93i)T2 1 + (-0.5 + 1.53i)T + (-4.04 - 2.93i)T^{2}
7 1+(−3.92+2.85i)T+(2.16−6.65i)T2 1 + (-3.92 + 2.85i)T + (2.16 - 6.65i)T^{2}
13 1+(−0.763−2.35i)T+(−10.5+7.64i)T2 1 + (-0.763 - 2.35i)T + (-10.5 + 7.64i)T^{2}
17 1+(−1.04+3.21i)T+(−13.7−9.99i)T2 1 + (-1.04 + 3.21i)T + (-13.7 - 9.99i)T^{2}
23 1+7.85T+23T2 1 + 7.85T + 23T^{2}
29 1+(8.47−6.15i)T+(8.96−27.5i)T2 1 + (8.47 - 6.15i)T + (8.96 - 27.5i)T^{2}
31 1+(−0.618−1.90i)T+(−25.0+18.2i)T2 1 + (-0.618 - 1.90i)T + (-25.0 + 18.2i)T^{2}
37 1+(3.23−2.35i)T+(11.4−35.1i)T2 1 + (3.23 - 2.35i)T + (11.4 - 35.1i)T^{2}
41 1+(−3.85−2.80i)T+(12.6+38.9i)T2 1 + (-3.85 - 2.80i)T + (12.6 + 38.9i)T^{2}
43 1+9.09T+43T2 1 + 9.09T + 43T^{2}
47 1+(1.11+0.812i)T+(14.5+44.6i)T2 1 + (1.11 + 0.812i)T + (14.5 + 44.6i)T^{2}
53 1+(1.47+4.53i)T+(−42.8+31.1i)T2 1 + (1.47 + 4.53i)T + (-42.8 + 31.1i)T^{2}
59 1+(−2.85+2.07i)T+(18.2−56.1i)T2 1 + (-2.85 + 2.07i)T + (18.2 - 56.1i)T^{2}
61 1+(2.95−9.09i)T+(−49.3−35.8i)T2 1 + (2.95 - 9.09i)T + (-49.3 - 35.8i)T^{2}
67 1−2.76T+67T2 1 - 2.76T + 67T^{2}
71 1+(−1.52+4.70i)T+(−57.4−41.7i)T2 1 + (-1.52 + 4.70i)T + (-57.4 - 41.7i)T^{2}
73 1+(−11.3+8.22i)T+(22.5−69.4i)T2 1 + (-11.3 + 8.22i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.14−3.52i)T+(−63.9+46.4i)T2 1 + (-1.14 - 3.52i)T + (-63.9 + 46.4i)T^{2}
83 1+(4.26−13.1i)T+(−67.1−48.7i)T2 1 + (4.26 - 13.1i)T + (-67.1 - 48.7i)T^{2}
89 1−11.2T+89T2 1 - 11.2T + 89T^{2}
97 1+(2.85+8.78i)T+(−78.4+57.0i)T2 1 + (2.85 + 8.78i)T + (-78.4 + 57.0i)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

−11.30838981935419160146676968256, −10.15221709446304125201640623812, −9.379243878701528732883628623813, −8.506918219679847992967217364815, −7.79033381213849726021429186433, −6.90799334665341064683220874788, −5.18921194857340329270528575339, −4.55977361686521505240658822173, −3.71011300161005432616438855563, −1.76828937003339794255804046403, 1.82940283674798730342952289288, 2.51051114997803138718570421131, 3.72469303087348185420765368918, 5.35201652597748484991408689131, 6.12086743482653085557087210220, 7.963888366366607151651566328179, 8.093781975588859108324934516506, 9.089695198523938625530104405380, 10.40521783690185483826907296948, 11.11667657109272319502761295426

Graph of the ZZ-function along the critical line