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Properties

Label 2-1216-152.27-c1-0-38
Degree 22
Conductor 12161216
Sign 0.0489−0.998i0.0489 - 0.998i
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−2.36 − 1.36i)5-s − 1.26i·7-s − 4.46·11-s + (2.36 + 4.09i)15-s + (2.73 − 4.73i)17-s + (−0.5 − 4.33i)19-s + (−1.09 + 1.90i)21-s + (−3.63 + 2.09i)23-s + (1.23 + 2.13i)25-s + 5.19i·27-s + (1.09 + 1.90i)29-s − 2.19·31-s + (6.69 + 3.86i)33-s + (−1.73 + 3.00i)35-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−1.05 − 0.610i)5-s − 0.479i·7-s − 1.34·11-s + (0.610 + 1.05i)15-s + (0.662 − 1.14i)17-s + (−0.114 − 0.993i)19-s + (−0.239 + 0.415i)21-s + (−0.757 + 0.437i)23-s + (0.246 + 0.426i)25-s + 0.999i·27-s + (0.203 + 0.353i)29-s − 0.394·31-s + (1.16 + 0.672i)33-s + (−0.292 + 0.507i)35-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=((0.0489−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=((0.0489−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: 0.0489−0.998i0.0489 - 0.998i
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1216(863,⋅)\chi_{1216} (863, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1216, ( :1/2), 0.0489−0.998i)(2,\ 1216,\ (\ :1/2),\ 0.0489 - 0.998i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
19 1+(0.5+4.33i)T 1 + (0.5 + 4.33i)T
good3 1+(1.5+0.866i)T+(1.5+2.59i)T2 1 + (1.5 + 0.866i)T + (1.5 + 2.59i)T^{2}
5 1+(2.36+1.36i)T+(2.5+4.33i)T2 1 + (2.36 + 1.36i)T + (2.5 + 4.33i)T^{2}
7 1+1.26iT−7T2 1 + 1.26iT - 7T^{2}
11 1+4.46T+11T2 1 + 4.46T + 11T^{2}
13 1+(−6.5+11.2i)T2 1 + (-6.5 + 11.2i)T^{2}
17 1+(−2.73+4.73i)T+(−8.5−14.7i)T2 1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2}
23 1+(3.63−2.09i)T+(11.5−19.9i)T2 1 + (3.63 - 2.09i)T + (11.5 - 19.9i)T^{2}
29 1+(−1.09−1.90i)T+(−14.5+25.1i)T2 1 + (-1.09 - 1.90i)T + (-14.5 + 25.1i)T^{2}
31 1+2.19T+31T2 1 + 2.19T + 31T^{2}
37 1+4.73T+37T2 1 + 4.73T + 37T^{2}
41 1+(−6.69−3.86i)T+(20.5+35.5i)T2 1 + (-6.69 - 3.86i)T + (20.5 + 35.5i)T^{2}
43 1+(−1.73+3i)T+(−21.5−37.2i)T2 1 + (-1.73 + 3i)T + (-21.5 - 37.2i)T^{2}
47 1+(8.36−4.83i)T+(23.5−40.7i)T2 1 + (8.36 - 4.83i)T + (23.5 - 40.7i)T^{2}
53 1+(−4.73−8.19i)T+(−26.5+45.8i)T2 1 + (-4.73 - 8.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.696−0.401i)T+(29.5+51.0i)T2 1 + (-0.696 - 0.401i)T + (29.5 + 51.0i)T^{2}
61 1+(−10.5+6.09i)T+(30.5−52.8i)T2 1 + (-10.5 + 6.09i)T + (30.5 - 52.8i)T^{2}
67 1+(−3.69+2.13i)T+(33.5−58.0i)T2 1 + (-3.69 + 2.13i)T + (33.5 - 58.0i)T^{2}
71 1+(1.26−2.19i)T+(−35.5−61.4i)T2 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2}
73 1+(−2.5+4.33i)T+(−36.5−63.2i)T2 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2}
79 1+(8.19−14.1i)T+(−39.5−68.4i)T2 1 + (8.19 - 14.1i)T + (-39.5 - 68.4i)T^{2}
83 1−3.39T+83T2 1 - 3.39T + 83T^{2}
89 1+(11.1−6.46i)T+(44.5−77.0i)T2 1 + (11.1 - 6.46i)T + (44.5 - 77.0i)T^{2}
97 1+(6.69+3.86i)T+(48.5+84.0i)T2 1 + (6.69 + 3.86i)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

−9.054594348370106354959594357225, −8.052420101715040672955909364341, −7.47764733621339253762428209427, −6.77523702346880101930218217442, −5.53950627704779727473066817125, −4.99830295761809621610591852870, −3.95040392874279238336211538598, −2.76398728565106356709405996052, −0.922295206936550637962057402849, 0, 2.25620591739892058766358954931, 3.49486431902112893523638122725, 4.31253683958934029031937233432, 5.47472899284056516914550446967, 5.88371764144996900914480826913, 7.09846534150064078162454146164, 8.064341166346484411623735593355, 8.362119013415235139609880166602, 9.975004536878591871849470234078

Graph of the ZZ-function along the critical line