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Properties

Label 2-2000-100.91-c0-0-1
Degree 22
Conductor 20002000
Sign 0.979+0.199i0.979 + 0.199i
Analytic cond. 0.9981300.998130
Root an. cond. 0.9990640.999064
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  + (−0.809 + 0.587i)9-s + (1.53 − 1.11i)13-s + (0.363 − 1.11i)17-s + (0.5 + 1.53i)29-s + (0.951 − 0.690i)37-s + (0.5 − 0.363i)41-s + 49-s + (0.587 + 1.80i)53-s + (0.5 + 0.363i)61-s + (−1.53 − 1.11i)73-s + (0.309 − 0.951i)81-s + (−1.30 − 0.951i)89-s + (−0.363 − 1.11i)97-s − 1.61·101-s + (−0.5 + 0.363i)109-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)9-s + (1.53 − 1.11i)13-s + (0.363 − 1.11i)17-s + (0.5 + 1.53i)29-s + (0.951 − 0.690i)37-s + (0.5 − 0.363i)41-s + 49-s + (0.587 + 1.80i)53-s + (0.5 + 0.363i)61-s + (−1.53 − 1.11i)73-s + (0.309 − 0.951i)81-s + (−1.30 − 0.951i)89-s + (−0.363 − 1.11i)97-s − 1.61·101-s + (−0.5 + 0.363i)109-s + ⋯

Functional equation

Λ(s)=(2000s/2 ΓC(s) L(s)=((0.979+0.199i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2000s/2 ΓC(s) L(s)=((0.979+0.199i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20002000    =    24⋅532^{4} \cdot 5^{3}
Sign: 0.979+0.199i0.979 + 0.199i
Analytic conductor: 0.9981300.998130
Root analytic conductor: 0.9990640.999064
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2000(1951,⋅)\chi_{2000} (1951, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2000, ( :0), 0.979+0.199i)(2,\ 2000,\ (\ :0),\ 0.979 + 0.199i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1805496101.180549610
L(12)L(\frac12) ≈\approx 1.1805496101.180549610
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
5 1 1
good3 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
7 1−T2 1 - T^{2}
11 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(−1.53+1.11i)T+(0.309−0.951i)T2 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2}
17 1+(−0.363+1.11i)T+(−0.809−0.587i)T2 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2}
19 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
23 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
29 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(−0.951+0.690i)T+(0.309−0.951i)T2 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2}
41 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
43 1−T2 1 - T^{2}
47 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
53 1+(−0.587−1.80i)T+(−0.809+0.587i)T2 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2}
59 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
61 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
67 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
71 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
73 1+(1.53+1.11i)T+(0.309+0.951i)T2 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2}
79 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
97 1+(0.363+1.11i)T+(−0.809+0.587i)T2 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)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

−9.087273974785495977407575684248, −8.627316926463935645053576225319, −7.79878855853348385993829684848, −7.09799888914971820749096978003, −5.86422127343905404335648622798, −5.56291498642464817665736578004, −4.47001950527944809916453376928, −3.31713973014315745940680911331, −2.64436361687078856065697842366, −1.07947189691273046784297102945, 1.24532942899974192960484432409, 2.53539253877544544433823084921, 3.71454461794880612354555124373, 4.23363318636722208178866935310, 5.61939153205642139382055891871, 6.20185062744357867077237942918, 6.80394513576726208222619503659, 8.145517420180831267814779765501, 8.456935787466045983137742980224, 9.321307489274425584849652293905

Graph of the ZZ-function along the critical line