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Properties

Label 2-3040-760.189-c0-0-3
Degree 22
Conductor 30403040
Sign 0.382−0.923i0.382 - 0.923i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + 0.765i·3-s − i·5-s + 0.414·9-s + 1.41i·11-s + 1.84i·13-s + 0.765·15-s − i·19-s − 25-s + 1.08i·27-s − 1.08·33-s − 0.765i·37-s − 1.41·39-s − 0.414i·45-s + 49-s + 0.765i·53-s + ⋯
L(s)  = 1  + 0.765i·3-s − i·5-s + 0.414·9-s + 1.41i·11-s + 1.84i·13-s + 0.765·15-s − i·19-s − 25-s + 1.08i·27-s − 1.08·33-s − 0.765i·37-s − 1.41·39-s − 0.414i·45-s + 49-s + 0.765i·53-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: 0.382−0.923i0.382 - 0.923i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(1329,⋅)\chi_{3040} (1329, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), 0.382−0.923i)(2,\ 3040,\ (\ :0),\ 0.382 - 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2543427201.254342720
L(12)L(\frac12) ≈\approx 1.2543427201.254342720
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+iT 1 + iT
19 1+iT 1 + iT
good3 1−0.765iT−T2 1 - 0.765iT - T^{2}
7 1−T2 1 - T^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1−1.84iT−T2 1 - 1.84iT - T^{2}
17 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+0.765iT−T2 1 + 0.765iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−T2 1 - T^{2}
53 1−0.765iT−T2 1 - 0.765iT - T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1−1.84iT−T2 1 - 1.84iT - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1−1.84T+T2 1 - 1.84T + 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

−9.129748814033673208581373831884, −8.637151050097844366503537064428, −7.26114097678814148834386054740, −7.09177978460601672414286812082, −5.85736233283713026343575168047, −4.84121552523362897590967111303, −4.43166052174304919375379569122, −3.93962004717039260350182385249, −2.34457006809601763802264094532, −1.45958532729500536077602201562, 0.835846983054331114201909965002, 2.14582435379029483008954824290, 3.18801086118079585636251968137, 3.65158255643870189561669688461, 5.10661096026082472519931247064, 6.02637728455404333475087028259, 6.35701216263126749130982312605, 7.38982409545786359463882144696, 7.924970340283411325417645668528, 8.411165588497022306453135833147

Graph of the ZZ-function along the critical line