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Properties

Label 2-1254-19.11-c1-0-11
Degree 22
Conductor 12541254
Sign −0.476−0.879i-0.476 - 0.879i
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.436 + 0.756i)5-s + (−0.499 − 0.866i)6-s + 2.30·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.436 − 0.756i)10-s + 11-s + 0.999·12-s + (−1.83 − 3.17i)13-s + (−1.15 + 1.99i)14-s + (−0.436 − 0.756i)15-s + (−0.5 + 0.866i)16-s + (−2.77 + 4.79i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.195 + 0.338i)5-s + (−0.204 − 0.353i)6-s + 0.870·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.138 − 0.239i)10-s + 0.301·11-s + 0.288·12-s + (−0.508 − 0.881i)13-s + (−0.307 + 0.533i)14-s + (−0.112 − 0.195i)15-s + (−0.125 + 0.216i)16-s + (−0.672 + 1.16i)17-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=((−0.476−0.879i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=((−0.476−0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: −0.476−0.879i-0.476 - 0.879i
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1254(1189,⋅)\chi_{1254} (1189, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), −0.476−0.879i)(2,\ 1254,\ (\ :1/2),\ -0.476 - 0.879i)

Particular Values

L(1)L(1) ≈\approx 1.1668718141.166871814
L(12)L(\frac12) ≈\approx 1.1668718141.166871814
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
11 1−T 1 - T
19 1+(−4.23−1.03i)T 1 + (-4.23 - 1.03i)T
good5 1+(0.436−0.756i)T+(−2.5−4.33i)T2 1 + (0.436 - 0.756i)T + (-2.5 - 4.33i)T^{2}
7 1−2.30T+7T2 1 - 2.30T + 7T^{2}
13 1+(1.83+3.17i)T+(−6.5+11.2i)T2 1 + (1.83 + 3.17i)T + (-6.5 + 11.2i)T^{2}
17 1+(2.77−4.79i)T+(−8.5−14.7i)T2 1 + (2.77 - 4.79i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.89+3.28i)T+(−11.5+19.9i)T2 1 + (1.89 + 3.28i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.41−7.64i)T+(−14.5+25.1i)T2 1 + (-4.41 - 7.64i)T + (-14.5 + 25.1i)T^{2}
31 1−8.71T+31T2 1 - 8.71T + 31T^{2}
37 1+1.49T+37T2 1 + 1.49T + 37T^{2}
41 1+(−2.77+4.81i)T+(−20.5−35.5i)T2 1 + (-2.77 + 4.81i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.0634+0.109i)T+(−21.5−37.2i)T2 1 + (-0.0634 + 0.109i)T + (-21.5 - 37.2i)T^{2}
47 1+(−2.22−3.86i)T+(−23.5+40.7i)T2 1 + (-2.22 - 3.86i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.29−2.24i)T+(−26.5+45.8i)T2 1 + (-1.29 - 2.24i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.15−7.19i)T+(−29.5−51.0i)T2 1 + (4.15 - 7.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.391−0.678i)T+(−30.5+52.8i)T2 1 + (-0.391 - 0.678i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.16+3.75i)T+(−33.5+58.0i)T2 1 + (2.16 + 3.75i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.79−10.0i)T+(−35.5−61.4i)T2 1 + (5.79 - 10.0i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.30−2.25i)T+(−36.5−63.2i)T2 1 + (1.30 - 2.25i)T + (-36.5 - 63.2i)T^{2}
79 1+(7.77−13.4i)T+(−39.5−68.4i)T2 1 + (7.77 - 13.4i)T + (-39.5 - 68.4i)T^{2}
83 1+10.2T+83T2 1 + 10.2T + 83T^{2}
89 1+(−4.69−8.13i)T+(−44.5+77.0i)T2 1 + (-4.69 - 8.13i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.84+6.66i)T+(−48.5−84.0i)T2 1 + (-3.84 + 6.66i)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

−10.12063570514000795019106736738, −8.968221857763368764727459873110, −8.359325808176145017148844365715, −7.55691901498429020810210695536, −6.70655034655967476274109250096, −5.77428367192249785784222976488, −4.96859012034291835806392943369, −4.15551674622343546962835983254, −2.88208773520331954159130395024, −1.23333601510744407328375194367, 0.67395903588777072062469894108, 1.86493820082782345531524465689, 2.88593477479503770510195898043, 4.48958876521336589694027345140, 4.80256821665096209033630333578, 6.18971431954536265152756735952, 7.15206159043781804287333645758, 7.87003326948656080697466217077, 8.632685003938375102203008159525, 9.467318284915433496960246118871

Graph of the ZZ-function along the critical line