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Properties

Label 2-30e2-5.2-c0-0-1
Degree 22
Conductor 900900
Sign 0.437+0.899i0.437 + 0.899i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
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  + (−1.22 − 1.22i)7-s + (1.22 − 1.22i)13-s − i·19-s + 31-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + (1.22 + 1.22i)67-s − 2i·79-s − 2.99·91-s + (1.22 + 1.22i)97-s + i·109-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)7-s + (1.22 − 1.22i)13-s − i·19-s + 31-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + (1.22 + 1.22i)67-s − 2i·79-s − 2.99·91-s + (1.22 + 1.22i)97-s + i·109-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.437+0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.437+0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.437+0.899i0.437 + 0.899i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(757,⋅)\chi_{900} (757, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.437+0.899i)(2,\ 900,\ (\ :0),\ 0.437 + 0.899i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87157069830.8715706983
L(12)L(\frac12) ≈\approx 0.87157069830.8715706983
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+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
17 1+iT2 1 + iT^{2}
19 1+iT−T2 1 + iT - T^{2}
23 1−iT2 1 - iT^{2}
29 1−T2 1 - T^{2}
31 1−T+T2 1 - T + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1+(1.22−1.22i)T−iT2 1 + (1.22 - 1.22i)T - iT^{2}
47 1+iT2 1 + iT^{2}
53 1−iT2 1 - iT^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
71 1+T2 1 + T^{2}
73 1−iT2 1 - iT^{2}
79 1+2iT−T2 1 + 2iT - T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{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

−10.25353791641766692345523681673, −9.462176123197861403004556759528, −8.459720517777055832355086295048, −7.59355858488529286103753863904, −6.66124527685326685047630787162, −6.05536351102163369485977570454, −4.76477946321248790789473686286, −3.65813058030188529823592112323, −2.95675114766430724629736550549, −0.913355882118608624809784294802, 1.87344142490513574483838728023, 3.13224511547547583092011411899, 4.02866988328401533974858982616, 5.40589337535647924101725424261, 6.26353907226557983656036270293, 6.73802011082313426482765853299, 8.170308511243535767782982734029, 8.862893281419158531774215734369, 9.533012411108889551851840406790, 10.32117757414847022994671968457

Graph of the ZZ-function along the critical line