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Properties

Label 2-40e2-5.3-c0-0-2
Degree 22
Conductor 16001600
Sign 0.525+0.850i0.525 + 0.850i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
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  − i·9-s + (−1 − i)13-s + (1 − i)17-s + (1 − i)37-s + i·49-s + (1 + i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s + 2·101-s − 2i·109-s + (1 + i)113-s + (−1 + i)117-s + ⋯
L(s)  = 1  − i·9-s + (−1 − i)13-s + (1 − i)17-s + (1 − i)37-s + i·49-s + (1 + i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s + 2·101-s − 2i·109-s + (1 + i)113-s + (−1 + i)117-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.525+0.850i0.525 + 0.850i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(193,⋅)\chi_{1600} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), 0.525+0.850i)(2,\ 1600,\ (\ :0),\ 0.525 + 0.850i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0395125451.039512545
L(12)L(\frac12) ≈\approx 1.0395125451.039512545
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+iT2 1 + iT^{2}
7 1−iT2 1 - iT^{2}
11 1+T2 1 + T^{2}
13 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
17 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
19 1−T2 1 - T^{2}
23 1+iT2 1 + iT^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1−iT2 1 - iT^{2}
53 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1+iT2 1 + iT^{2}
89 1−T2 1 - T^{2}
97 1+(1−i)T−iT2 1 + (1 - i)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

−9.548399693725478716586461596171, −8.798436193317607234406669320793, −7.66579803539722451925421093393, −7.29226262087773670281017124505, −6.13321397127507000793770628340, −5.44700313966039662178670357305, −4.48220157498074805728549511348, −3.36643280550921688563021665879, −2.58330657180352164610614307723, −0.860198330781881761685279441821, 1.66118871882881372317107720211, 2.64759275157110111882515957514, 3.90170104291048891170229486744, 4.78836929313519103417948617163, 5.56364618365182883556818772206, 6.56712806286335577060941872781, 7.41883908849089870645206731706, 8.091247731466074846928739283111, 8.876564417090761194257583351309, 9.961734970081460408075153894472

Graph of the ZZ-function along the critical line