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Properties

Label 2-315-21.20-c9-0-42
Degree 22
Conductor 315315
Sign 0.102+0.994i0.102 + 0.994i
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  − 38.5i·2-s − 971.·4-s + 625·5-s + (5.53e3 + 3.11e3i)7-s + 1.77e4i·8-s − 2.40e4i·10-s + 3.58e4i·11-s − 1.02e5i·13-s + (1.20e5 − 2.13e5i)14-s + 1.84e5·16-s + 2.29e5·17-s − 4.84e5i·19-s − 6.07e5·20-s + 1.38e6·22-s + 6.86e5i·23-s + ⋯
L(s)  = 1  − 1.70i·2-s − 1.89·4-s + 0.447·5-s + (0.871 + 0.490i)7-s + 1.52i·8-s − 0.761i·10-s + 0.738i·11-s − 1.00i·13-s + (0.835 − 1.48i)14-s + 0.705·16-s + 0.667·17-s − 0.852i·19-s − 0.848·20-s + 1.25·22-s + 0.511i·23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((0.102+0.994i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+9/2) L(s)=((0.102+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 0.102+0.994i0.102 + 0.994i
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), 0.102+0.994i)(2,\ 315,\ (\ :9/2),\ 0.102 + 0.994i)

Particular Values

L(5)L(5) ≈\approx 2.3712869972.371286997
L(12)L(\frac12) ≈\approx 2.3712869972.371286997
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+(−5.53e3−3.11e3i)T 1 + (-5.53e3 - 3.11e3i)T
good2 1+38.5iT−512T2 1 + 38.5iT - 512T^{2}
11 1−3.58e4iT−2.35e9T2 1 - 3.58e4iT - 2.35e9T^{2}
13 1+1.02e5iT−1.06e10T2 1 + 1.02e5iT - 1.06e10T^{2}
17 1−2.29e5T+1.18e11T2 1 - 2.29e5T + 1.18e11T^{2}
19 1+4.84e5iT−3.22e11T2 1 + 4.84e5iT - 3.22e11T^{2}
23 1−6.86e5iT−1.80e12T2 1 - 6.86e5iT - 1.80e12T^{2}
29 1−2.83e6iT−1.45e13T2 1 - 2.83e6iT - 1.45e13T^{2}
31 1−2.96e6iT−2.64e13T2 1 - 2.96e6iT - 2.64e13T^{2}
37 1+1.64e7T+1.29e14T2 1 + 1.64e7T + 1.29e14T^{2}
41 1+1.37e7T+3.27e14T2 1 + 1.37e7T + 3.27e14T^{2}
43 1−1.57e7T+5.02e14T2 1 - 1.57e7T + 5.02e14T^{2}
47 1−2.20e7T+1.11e15T2 1 - 2.20e7T + 1.11e15T^{2}
53 1+5.73e7iT−3.29e15T2 1 + 5.73e7iT - 3.29e15T^{2}
59 1−2.98e7T+8.66e15T2 1 - 2.98e7T + 8.66e15T^{2}
61 1−2.12e8iT−1.16e16T2 1 - 2.12e8iT - 1.16e16T^{2}
67 1−1.05e8T+2.72e16T2 1 - 1.05e8T + 2.72e16T^{2}
71 1+1.24e8iT−4.58e16T2 1 + 1.24e8iT - 4.58e16T^{2}
73 1−2.55e8iT−5.88e16T2 1 - 2.55e8iT - 5.88e16T^{2}
79 1−2.67e8T+1.19e17T2 1 - 2.67e8T + 1.19e17T^{2}
83 1+1.62e8T+1.86e17T2 1 + 1.62e8T + 1.86e17T^{2}
89 1−7.43e8T+3.50e17T2 1 - 7.43e8T + 3.50e17T^{2}
97 1−4.95e8iT−7.60e17T2 1 - 4.95e8iT - 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

−10.18885112059590775438427041695, −9.218756210812206099690016817622, −8.439685957182477746545269714005, −7.17664785418072721357667174647, −5.44594025459696092739526175524, −4.83722748177342493068835648676, −3.52887783974627961781875822687, −2.55410709127012544809458994702, −1.70266195000373315928197998298, −0.835573627598076713625921363873, 0.57883155196224265992619547019, 1.89304964127468880341454734600, 3.77863896646317023136021454396, 4.76980448304055216812849689468, 5.67933827082066483169635069544, 6.47160611563778908000823095921, 7.44994197870128616483144567906, 8.212604192504959318699445740468, 8.985249635416960198321912131714, 10.03849842605192585883297545078

Graph of the ZZ-function along the critical line