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Properties

Label 2-2496-312.275-c0-0-0
Degree 22
Conductor 24962496
Sign −0.233−0.972i-0.233 - 0.972i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (0.133 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.366 + 1.36i)19-s + (−0.366 + 0.366i)21-s + i·25-s − 0.999·27-s + (1.36 + 1.36i)31-s + (−1.36 − 0.366i)37-s − 0.999i·39-s + (0.866 + 0.5i)43-s + (0.633 − 0.366i)49-s + (−0.999 + i)57-s + (−0.866 − 0.5i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.133 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.366 + 1.36i)19-s + (−0.366 + 0.366i)21-s + i·25-s − 0.999·27-s + (1.36 + 1.36i)31-s + (−1.36 − 0.366i)37-s − 0.999i·39-s + (0.866 + 0.5i)43-s + (0.633 − 0.366i)49-s + (−0.999 + i)57-s + (−0.866 − 0.5i)61-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=((−0.233−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s) L(s)=((−0.233−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: −0.233−0.972i-0.233 - 0.972i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2496(1055,⋅)\chi_{2496} (1055, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2496, ( :0), −0.233−0.972i)(2,\ 2496,\ (\ :0),\ -0.233 - 0.972i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2872446081.287244608
L(12)L(\frac12) ≈\approx 1.2872446081.287244608
L(1)L(1) 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
3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good5 1−iT2 1 - iT^{2}
7 1+(−0.133−0.5i)T+(−0.866+0.5i)T2 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2}
11 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+(−1.36−1.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
37 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
41 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
43 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
47 1+iT2 1 + iT^{2}
53 1+T2 1 + T^{2}
59 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
61 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
67 1+(0.5+0.133i)T+(0.866+0.5i)T2 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2}
71 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
73 1+(−0.366−0.366i)T+iT2 1 + (-0.366 - 0.366i)T + iT^{2}
79 1+1.73iT−T2 1 + 1.73iT - T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
97 1+(0.5+1.86i)T+(−0.866+0.5i)T2 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)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

−9.260710298729083020265766884949, −8.680191453647401546587958871600, −7.906326279459783654165073306257, −7.25982427768462984186911939304, −6.01270788494567949783661299101, −5.28627355015494738689279689123, −4.65010498193238132387826834744, −3.54871748142952499290944302511, −2.89403190274989759179588307267, −1.77484794444131334455209268037, 0.804647898068817346502806250613, 2.18236534610030329694531345954, 2.85900411752761296474430016003, 4.06767484707079852693662532683, 4.84633484350411841906878057583, 5.97983267166276215685742683831, 6.80970520504917519731923375155, 7.31246798381799790775470463195, 8.032652344164921161942164488748, 8.828114123858920054868142065778

Graph of the ZZ-function along the critical line