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Properties

Label 2-2700-15.14-c0-0-0
Degree 22
Conductor 27002700
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 1.347471.34747
Root an. cond. 1.160801.16080
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  + 2i·7-s + i·13-s − 2·19-s − 31-s − i·37-s + i·43-s − 3·49-s + 2·61-s − i·67-s + i·73-s + 79-s − 2·91-s + 2i·97-s + i·103-s + 109-s + ⋯
L(s)  = 1  + 2i·7-s + i·13-s − 2·19-s − 31-s − i·37-s + i·43-s − 3·49-s + 2·61-s − i·67-s + i·73-s + 79-s − 2·91-s + 2i·97-s + i·103-s + 109-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 1.347471.34747
Root analytic conductor: 1.160801.16080
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2700(1349,⋅)\chi_{2700} (1349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :0), −0.447−0.894i)(2,\ 2700,\ (\ :0),\ -0.447 - 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92207216390.9220721639
L(12)L(\frac12) ≈\approx 0.92207216390.9220721639
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 1
5 1 1
good7 1−2iT−T2 1 - 2iT - T^{2}
11 1−T2 1 - T^{2}
13 1−iT−T2 1 - iT - T^{2}
17 1+T2 1 + T^{2}
19 1+2T+T2 1 + 2T + T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1+T+T2 1 + T + T^{2}
37 1+iT−T2 1 + iT - T^{2}
41 1−T2 1 - T^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−2T+T2 1 - 2T + T^{2}
67 1+iT−T2 1 + iT - T^{2}
71 1−T2 1 - T^{2}
73 1−iT−T2 1 - iT - T^{2}
79 1−T+T2 1 - T + T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1−2iT−T2 1 - 2iT - 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

−9.091637785423212198883693756993, −8.681202808432108185843661024514, −7.963934002418641488404220151691, −6.77359256091689445132605530201, −6.20626766168947538560534306563, −5.47505686787301888731190198415, −4.64136848189099960822936788501, −3.66752309616216248150560617331, −2.39021057010672013546513966627, −1.98235594742285009280200967455, 0.56401318666265424352534726049, 1.88673787759696423020837282265, 3.25534057957201705989848958104, 4.01643959780259350109852983031, 4.66056966403181168801545658913, 5.72113353838417469636715861883, 6.70740244205642144942380433275, 7.17440219770292500724061322933, 8.037499074387559441106608185730, 8.572446661602938616566236442309

Graph of the ZZ-function along the critical line