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Properties

Label 2-3920-980.799-c0-0-1
Degree 22
Conductor 39203920
Sign 0.981+0.191i0.981 + 0.191i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  + (−0.974 + 1.22i)3-s + (−0.623 + 0.781i)5-s + (0.433 − 0.900i)7-s + (−0.321 − 1.40i)9-s + (−0.347 − 1.52i)15-s + (0.678 + 1.40i)21-s + (−1.40 − 0.678i)23-s + (−0.222 − 0.974i)25-s + (0.626 + 0.301i)27-s + (1.62 − 0.781i)29-s + (0.433 + 0.900i)35-s + (0.277 − 0.347i)41-s + (−1.21 − 1.52i)43-s + (1.30 + 0.626i)45-s + (0.347 − 1.52i)47-s + ⋯
L(s)  = 1  + (−0.974 + 1.22i)3-s + (−0.623 + 0.781i)5-s + (0.433 − 0.900i)7-s + (−0.321 − 1.40i)9-s + (−0.347 − 1.52i)15-s + (0.678 + 1.40i)21-s + (−1.40 − 0.678i)23-s + (−0.222 − 0.974i)25-s + (0.626 + 0.301i)27-s + (1.62 − 0.781i)29-s + (0.433 + 0.900i)35-s + (0.277 − 0.347i)41-s + (−1.21 − 1.52i)43-s + (1.30 + 0.626i)45-s + (0.347 − 1.52i)47-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((0.981+0.191i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((0.981+0.191i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: 0.981+0.191i0.981 + 0.191i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(799,⋅)\chi_{3920} (799, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), 0.981+0.191i)(2,\ 3920,\ (\ :0),\ 0.981 + 0.191i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62376099370.6237609937
L(12)L(\frac12) ≈\approx 0.62376099370.6237609937
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+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
7 1+(−0.433+0.900i)T 1 + (-0.433 + 0.900i)T
good3 1+(0.974−1.22i)T+(−0.222−0.974i)T2 1 + (0.974 - 1.22i)T + (-0.222 - 0.974i)T^{2}
11 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
13 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
17 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
19 1−T2 1 - T^{2}
23 1+(1.40+0.678i)T+(0.623+0.781i)T2 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2}
29 1+(−1.62+0.781i)T+(0.623−0.781i)T2 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
41 1+(−0.277+0.347i)T+(−0.222−0.974i)T2 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2}
43 1+(1.21+1.52i)T+(−0.222+0.974i)T2 1 + (1.21 + 1.52i)T + (-0.222 + 0.974i)T^{2}
47 1+(−0.347+1.52i)T+(−0.900−0.433i)T2 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2}
53 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
59 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
61 1+(−1.12+0.541i)T+(0.623−0.781i)T2 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
73 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
79 1−T2 1 - T^{2}
83 1+(−0.193−0.846i)T+(−0.900+0.433i)T2 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2}
89 1+(−0.400−1.75i)T+(−0.900+0.433i)T2 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2}
97 1−T2 1 - 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

−8.477169841537834399042195444902, −7.988419467422051236790682121149, −6.94040745041350949054795038920, −6.47699648102777439067894284836, −5.52025202040431117190535326329, −4.73542750242663310351275015107, −4.02872254450270482330203266653, −3.63588560807637010403818997973, −2.30300176695711612860100726426, −0.46711404657144153919783925067, 1.10618767698904458490661547983, 1.86851935791526856574090796957, 3.06476326096413830901028466821, 4.40113764083080741905629171400, 5.04221771838932022024704308408, 5.81903374297771173928634429675, 6.33005867373873858394755790568, 7.24904326011572256949487591963, 7.970628030916402508102586313574, 8.358785415650033458831048133640

Graph of the ZZ-function along the critical line