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Properties

Label 2-315-21.20-c9-0-31
Degree 22
Conductor 315315
Sign −0.997−0.0722i-0.997 - 0.0722i
Analytic cond. 162.236162.236
Root an. cond. 12.737212.7372
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.2i·2-s + 246.·4-s + 625·5-s + (−4.03e3 + 4.90e3i)7-s + 1.23e4i·8-s + 1.01e4i·10-s + 7.41e4i·11-s − 1.88e5i·13-s + (−7.99e4 − 6.56e4i)14-s − 7.47e4·16-s + 4.57e5·17-s + 5.34e5i·19-s + 1.54e5·20-s − 1.20e6·22-s + 8.58e5i·23-s + ⋯
L(s)  = 1  + 0.719i·2-s + 0.482·4-s + 0.447·5-s + (−0.634 + 0.772i)7-s + 1.06i·8-s + 0.321i·10-s + 1.52i·11-s − 1.83i·13-s + (−0.556 − 0.456i)14-s − 0.285·16-s + 1.32·17-s + 0.940i·19-s + 0.215·20-s − 1.09·22-s + 0.639i·23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((−0.997−0.0722i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+9/2) L(s)=((−0.997−0.0722i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −0.997−0.0722i-0.997 - 0.0722i
Analytic conductor: 162.236162.236
Root analytic conductor: 12.737212.7372
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ315(251,⋅)\chi_{315} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 315, ( :9/2), −0.997−0.0722i)(2,\ 315,\ (\ :9/2),\ -0.997 - 0.0722i)

Particular Values

L(5)L(5) ≈\approx 2.5071572742.507157274
L(12)L(\frac12) ≈\approx 2.5071572742.507157274
L(112)L(\frac{11}{2}) 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)
bad3 1 1
5 1−625T 1 - 625T
7 1+(4.03e3−4.90e3i)T 1 + (4.03e3 - 4.90e3i)T
good2 1−16.2iT−512T2 1 - 16.2iT - 512T^{2}
11 1−7.41e4iT−2.35e9T2 1 - 7.41e4iT - 2.35e9T^{2}
13 1+1.88e5iT−1.06e10T2 1 + 1.88e5iT - 1.06e10T^{2}
17 1−4.57e5T+1.18e11T2 1 - 4.57e5T + 1.18e11T^{2}
19 1−5.34e5iT−3.22e11T2 1 - 5.34e5iT - 3.22e11T^{2}
23 1−8.58e5iT−1.80e12T2 1 - 8.58e5iT - 1.80e12T^{2}
29 1−4.72e6iT−1.45e13T2 1 - 4.72e6iT - 1.45e13T^{2}
31 1−2.47e5iT−2.64e13T2 1 - 2.47e5iT - 2.64e13T^{2}
37 1−1.57e7T+1.29e14T2 1 - 1.57e7T + 1.29e14T^{2}
41 1+1.50e7T+3.27e14T2 1 + 1.50e7T + 3.27e14T^{2}
43 1−4.15e6T+5.02e14T2 1 - 4.15e6T + 5.02e14T^{2}
47 1+1.30e7T+1.11e15T2 1 + 1.30e7T + 1.11e15T^{2}
53 1+9.25e7iT−3.29e15T2 1 + 9.25e7iT - 3.29e15T^{2}
59 1−1.12e8T+8.66e15T2 1 - 1.12e8T + 8.66e15T^{2}
61 1+3.91e7iT−1.16e16T2 1 + 3.91e7iT - 1.16e16T^{2}
67 1−3.12e6T+2.72e16T2 1 - 3.12e6T + 2.72e16T^{2}
71 1−3.51e8iT−4.58e16T2 1 - 3.51e8iT - 4.58e16T^{2}
73 1−4.73e8iT−5.88e16T2 1 - 4.73e8iT - 5.88e16T^{2}
79 1+3.62e8T+1.19e17T2 1 + 3.62e8T + 1.19e17T^{2}
83 1+2.56e8T+1.86e17T2 1 + 2.56e8T + 1.86e17T^{2}
89 1−7.32e8T+3.50e17T2 1 - 7.32e8T + 3.50e17T^{2}
97 1+2.33e8iT−7.60e17T2 1 + 2.33e8iT - 7.60e17T^{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

−10.14471516472393140718120188782, −9.873990463128710952078723463920, −8.412477134349977546943855874851, −7.63119292742937022006848996882, −6.74524648077229300560143729261, −5.61192559520140008683345926448, −5.30650312979133634960231573400, −3.37357298102064781164578246180, −2.45036850243422958145537490433, −1.33686365042867680018281314550, 0.45923721410385796047079988224, 1.25800571251061845940148500776, 2.49875105321338939995313811091, 3.38578850722041178655770119295, 4.35467374873729417454261812019, 6.03647217165897199244409674808, 6.59003200469797116757978803884, 7.64906143520517208804001337968, 9.044568049559795365344686475803, 9.789291157330055211239984523169

Graph of the ZZ-function along the critical line