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Properties

Label 2-2100-105.62-c0-0-2
Degree 22
Conductor 21002100
Sign 0.0706−0.997i0.0706 - 0.997i
Analytic cond. 1.048031.04803
Root an. cond. 1.023731.02373
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.707 + 0.707i)13-s − 1.73·19-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s − 1.73i·31-s + 1.00i·39-s + (1.22 − 1.22i)43-s + (−0.866 + 0.499i)49-s + (−1.22 − 1.22i)57-s + 1.73i·61-s + (−0.965 + 0.258i)63-s + (1.22 + 1.22i)67-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.707 + 0.707i)13-s − 1.73·19-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s − 1.73i·31-s + 1.00i·39-s + (1.22 − 1.22i)43-s + (−0.866 + 0.499i)49-s + (−1.22 − 1.22i)57-s + 1.73i·61-s + (−0.965 + 0.258i)63-s + (1.22 + 1.22i)67-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((0.0706−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s) L(s)=((0.0706−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: 0.0706−0.997i0.0706 - 0.997i
Analytic conductor: 1.048031.04803
Root analytic conductor: 1.023731.02373
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2100(2057,⋅)\chi_{2100} (2057, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :0), 0.0706−0.997i)(2,\ 2100,\ (\ :0),\ 0.0706 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4782632681.478263268
L(12)L(\frac12) ≈\approx 1.4782632681.478263268
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1 1
7 1+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
good11 1−T2 1 - T^{2}
13 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
17 1+iT2 1 + iT^{2}
19 1+1.73T+T2 1 + 1.73T + T^{2}
23 1+iT2 1 + iT^{2}
29 1+T2 1 + T^{2}
31 1+1.73iT−T2 1 + 1.73iT - T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
47 1+iT2 1 + iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1−1.73iT−T2 1 - 1.73iT - T^{2}
67 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
71 1−T2 1 - T^{2}
73 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
79 1+2iT−T2 1 + 2iT - T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{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

−9.314367506554107893165909924810, −8.684351906742652740344929119883, −8.293475794220688867863316582286, −7.26866737789852413349185359918, −6.18661038426836517400163536251, −5.51416036628367077679939570871, −4.38684835536315833433932550365, −3.90742980170201014452337436259, −2.59860272279242176024236635335, −1.96360172655586666706142522880, 1.01380578113659531239758632207, 2.12040166946420337732750179844, 3.28834153092104704587477894295, 4.00095611108741010692426345730, 5.02761707728075642182977994399, 6.33827192082420388244126585189, 6.69596310019873133251717065592, 7.76944351024615759340716469239, 8.173690481748484119114611249295, 8.921161231531229771691319567445

Graph of the ZZ-function along the critical line