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Properties

Label 2-48-48.11-c1-0-0
Degree 22
Conductor 4848
Sign 0.176−0.984i0.176 - 0.984i
Analytic cond. 0.3832810.383281
Root an. cond. 0.6190970.619097
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  + (0.416 + 1.35i)2-s + (−1.43 + 0.966i)3-s + (−1.65 + 1.12i)4-s + (1.57 − 1.57i)5-s + (−1.90 − 1.54i)6-s + 2.24·7-s + (−2.20 − 1.76i)8-s + (1.13 − 2.77i)9-s + (2.77 + 1.47i)10-s + (−1.13 − 1.13i)11-s + (1.29 − 3.21i)12-s + (−3.24 + 3.24i)13-s + (0.935 + 3.04i)14-s + (−0.739 + 3.77i)15-s + (1.47 − 3.71i)16-s − 1.66i·17-s + ⋯
L(s)  = 1  + (0.294 + 0.955i)2-s + (−0.829 + 0.558i)3-s + (−0.826 + 0.562i)4-s + (0.702 − 0.702i)5-s + (−0.777 − 0.628i)6-s + 0.850·7-s + (−0.780 − 0.624i)8-s + (0.377 − 0.926i)9-s + (0.878 + 0.465i)10-s + (−0.341 − 0.341i)11-s + (0.372 − 0.928i)12-s + (−0.901 + 0.901i)13-s + (0.250 + 0.812i)14-s + (−0.191 + 0.975i)15-s + (0.367 − 0.929i)16-s − 0.403i·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.176−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1/2) L(s)=((0.176−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.176−0.984i0.176 - 0.984i
Analytic conductor: 0.3832810.383281
Root analytic conductor: 0.6190970.619097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ48(11,⋅)\chi_{48} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :1/2), 0.176−0.984i)(2,\ 48,\ (\ :1/2),\ 0.176 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 0.599699+0.501865i0.599699 + 0.501865i
L(12)L(\frac12) ≈\approx 0.599699+0.501865i0.599699 + 0.501865i
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+(−0.416−1.35i)T 1 + (-0.416 - 1.35i)T
3 1+(1.43−0.966i)T 1 + (1.43 - 0.966i)T
good5 1+(−1.57+1.57i)T−5iT2 1 + (-1.57 + 1.57i)T - 5iT^{2}
7 1−2.24T+7T2 1 - 2.24T + 7T^{2}
11 1+(1.13+1.13i)T+11iT2 1 + (1.13 + 1.13i)T + 11iT^{2}
13 1+(3.24−3.24i)T−13iT2 1 + (3.24 - 3.24i)T - 13iT^{2}
17 1+1.66iT−17T2 1 + 1.66iT - 17T^{2}
19 1+(3.77+3.77i)T+19iT2 1 + (3.77 + 3.77i)T + 19iT^{2}
23 1−2.26iT−23T2 1 - 2.26iT - 23T^{2}
29 1+(−3.23−3.23i)T+29iT2 1 + (-3.23 - 3.23i)T + 29iT^{2}
31 1+1.30iT−31T2 1 + 1.30iT - 31T^{2}
37 1+(−2.30−2.30i)T+37iT2 1 + (-2.30 - 2.30i)T + 37iT^{2}
41 1+10.2T+41T2 1 + 10.2T + 41T^{2}
43 1+(−3.77+3.77i)T−43iT2 1 + (-3.77 + 3.77i)T - 43iT^{2}
47 1−3.74T+47T2 1 - 3.74T + 47T^{2}
53 1+(0.972−0.972i)T−53iT2 1 + (0.972 - 0.972i)T - 53iT^{2}
59 1+(3.88+3.88i)T+59iT2 1 + (3.88 + 3.88i)T + 59iT^{2}
61 1+(−4.19+4.19i)T−61iT2 1 + (-4.19 + 4.19i)T - 61iT^{2}
67 1+(−8.02−8.02i)T+67iT2 1 + (-8.02 - 8.02i)T + 67iT^{2}
71 1−11.0iT−71T2 1 - 11.0iT - 71T^{2}
73 1−6.38iT−73T2 1 - 6.38iT - 73T^{2}
79 1+2.69iT−79T2 1 + 2.69iT - 79T^{2}
83 1+(2.61−2.61i)T−83iT2 1 + (2.61 - 2.61i)T - 83iT^{2}
89 1−7.35T+89T2 1 - 7.35T + 89T^{2}
97 1+5.67T+97T2 1 + 5.67T + 97T^{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

−16.06076839313560804303343518104, −14.97890021226482603588154770718, −13.85157533333164765927770932037, −12.64102932762574931537437456828, −11.42234074808702762056818484713, −9.752725871091366094722358297328, −8.681104814178847815338033007124, −6.89700492669798142989768056944, −5.39287300666777092378821769418, −4.60154615981983080022896872675, 2.19759953500604809690517725123, 4.86923829016358286536353615561, 6.16292571127716224411723999073, 8.008315207075058814838626075355, 10.16134878515106964803907815814, 10.68776409511226974382761045146, 12.01781200717581045087508472205, 12.89197755385284485518647476138, 14.10739171418719838041759868407, 15.03107225786769395318045495470

Graph of the ZZ-function along the critical line