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Properties

Label 2-1254-209.164-c1-0-4
Degree 22
Conductor 12541254
Sign −0.989+0.144i-0.989 + 0.144i
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
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 + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (1.37 + 2.38i)5-s + (−0.866 − 0.499i)6-s − 4.25i·7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−1.37 + 2.38i)10-s + (1.41 − 2.99i)11-s − 0.999i·12-s + (−3.47 + 6.01i)13-s + (3.68 − 2.12i)14-s + (−2.38 − 1.37i)15-s + (−0.5 − 0.866i)16-s + (−4.76 + 2.75i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.616 + 1.06i)5-s + (−0.353 − 0.204i)6-s − 1.60i·7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.436 + 0.755i)10-s + (0.428 − 0.903i)11-s − 0.288i·12-s + (−0.963 + 1.66i)13-s + (0.984 − 0.568i)14-s + (−0.616 − 0.356i)15-s + (−0.125 − 0.216i)16-s + (−1.15 + 0.667i)17-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=((−0.989+0.144i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=((−0.989+0.144i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: −0.989+0.144i-0.989 + 0.144i
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1254(373,⋅)\chi_{1254} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), −0.989+0.144i)(2,\ 1254,\ (\ :1/2),\ -0.989 + 0.144i)

Particular Values

L(1)L(1) ≈\approx 0.91296016830.9129601683
L(12)L(\frac12) ≈\approx 0.91296016830.9129601683
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
11 1+(−1.41+2.99i)T 1 + (-1.41 + 2.99i)T
19 1+(4.30−0.701i)T 1 + (4.30 - 0.701i)T
good5 1+(−1.37−2.38i)T+(−2.5+4.33i)T2 1 + (-1.37 - 2.38i)T + (-2.5 + 4.33i)T^{2}
7 1+4.25iT−7T2 1 + 4.25iT - 7T^{2}
13 1+(3.47−6.01i)T+(−6.5−11.2i)T2 1 + (3.47 - 6.01i)T + (-6.5 - 11.2i)T^{2}
17 1+(4.76−2.75i)T+(8.5−14.7i)T2 1 + (4.76 - 2.75i)T + (8.5 - 14.7i)T^{2}
23 1+(0.882−1.52i)T+(−11.5−19.9i)T2 1 + (0.882 - 1.52i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.07−7.05i)T+(−14.5−25.1i)T2 1 + (4.07 - 7.05i)T + (-14.5 - 25.1i)T^{2}
31 1−9.97iT−31T2 1 - 9.97iT - 31T^{2}
37 1−7.57iT−37T2 1 - 7.57iT - 37T^{2}
41 1+(4.84+8.39i)T+(−20.5+35.5i)T2 1 + (4.84 + 8.39i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.39−1.38i)T+(21.5−37.2i)T2 1 + (2.39 - 1.38i)T + (21.5 - 37.2i)T^{2}
47 1+(−2.20+3.82i)T+(−23.5−40.7i)T2 1 + (-2.20 + 3.82i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.984−0.568i)T+(26.5+45.8i)T2 1 + (-0.984 - 0.568i)T + (26.5 + 45.8i)T^{2}
59 1+(2.65−1.53i)T+(29.5−51.0i)T2 1 + (2.65 - 1.53i)T + (29.5 - 51.0i)T^{2}
61 1+(−11.1−6.42i)T+(30.5+52.8i)T2 1 + (-11.1 - 6.42i)T + (30.5 + 52.8i)T^{2}
67 1+(10.9+6.33i)T+(33.5+58.0i)T2 1 + (10.9 + 6.33i)T + (33.5 + 58.0i)T^{2}
71 1+(−4.40+2.54i)T+(35.5−61.4i)T2 1 + (-4.40 + 2.54i)T + (35.5 - 61.4i)T^{2}
73 1+(−11.1+6.44i)T+(36.5−63.2i)T2 1 + (-11.1 + 6.44i)T + (36.5 - 63.2i)T^{2}
79 1+(−2.97−5.15i)T+(−39.5+68.4i)T2 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2}
83 1+0.905iT−83T2 1 + 0.905iT - 83T^{2}
89 1+(−1.05−0.611i)T+(44.5+77.0i)T2 1 + (-1.05 - 0.611i)T + (44.5 + 77.0i)T^{2}
97 1+(1.40−0.812i)T+(48.5−84.0i)T2 1 + (1.40 - 0.812i)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

−10.41717827535281218882915997708, −9.310976285287791585999536106386, −8.505663343111697291214898151851, −7.04187704850927400612589989430, −6.85767728187999062728156192960, −6.28721758043976741192572348187, −5.00291537834277050857006363413, −4.14925301142443574858900609499, −3.42996713135243143183525881842, −1.82486052103613800025461970110, 0.33679317998574431107465008220, 2.09366429212032356962769614245, 2.45174613173152441414172513208, 4.32696395589164028676325436364, 5.09416267746237920998709848900, 5.65311721548248203736127285276, 6.42023877168418437257718035828, 7.76356280462216042656148619501, 8.666285721478665415571742056069, 9.491482357568476707427523526569

Graph of the ZZ-function along the critical line