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Properties

Label 2-2100-21.11-c0-0-2
Degree 22
Conductor 21002100
Sign −0.895+0.444i-0.895 + 0.444i
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  + (0.5 − 0.866i)3-s − 7-s + (−0.499 − 0.866i)9-s − 2·13-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)21-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + 43-s + 49-s − 1.99·57-s + (0.5 + 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s − 7-s + (−0.499 − 0.866i)9-s − 2·13-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)21-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + 43-s + 49-s − 1.99·57-s + (0.5 + 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((−0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s) L(s)=((−0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: −0.895+0.444i-0.895 + 0.444i
Analytic conductor: 1.048031.04803
Root analytic conductor: 1.023731.02373
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2100(1901,⋅)\chi_{2100} (1901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :0), −0.895+0.444i)(2,\ 2100,\ (\ :0),\ -0.895 + 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69821057710.6982105771
L(12)L(\frac12) ≈\approx 0.69821057710.6982105771
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
5 1 1
7 1+T 1 + T
good11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+2T+T2 1 + 2T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1−T+T2 1 - T + 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

−9.119067987565517800567991667680, −8.104620255449959002183226012153, −7.21903574354301994920824613477, −6.87661785654432071081895943124, −6.01035895553357670349764099437, −4.95326137885336850924624301160, −3.92450277127397681537746845221, −2.66605004214087088489473630842, −2.35118307705823380121013576884, −0.41741284568410749007367516442, 2.13866151895462344306578545279, 2.98978676491834756275044514366, 3.86823952812899062202965003619, 4.70786124998326249892450902016, 5.54607648150455333156157480689, 6.48948922826676185148570891508, 7.39306049119319320721627673730, 8.176069391337383464268993776714, 8.957068056411632745930208733391, 9.827936326910205045853000461930

Graph of the ZZ-function along the critical line