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Properties

Label 2-175-5.4-c9-0-53
Degree 22
Conductor 175175
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 90.131290.1312
Root an. cond. 9.493749.49374
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  − 14.8i·2-s + 239. i·3-s + 292.·4-s + 3.55e3·6-s + 2.40e3i·7-s − 1.19e4i·8-s − 3.77e4·9-s − 1.65e4·11-s + 7.00e4i·12-s − 2.63e4i·13-s + 3.56e4·14-s − 2.72e4·16-s − 1.44e5i·17-s + 5.60e5i·18-s + 1.59e5·19-s + ⋯
L(s)  = 1  − 0.655i·2-s + 1.70i·3-s + 0.570·4-s + 1.11·6-s + 0.377i·7-s − 1.02i·8-s − 1.91·9-s − 0.340·11-s + 0.974i·12-s − 0.255i·13-s + 0.247·14-s − 0.103·16-s − 0.418i·17-s + 1.25i·18-s + 0.281·19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+9/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 90.131290.1312
Root analytic conductor: 9.493749.49374
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ175(99,⋅)\chi_{175} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :9/2), 0.894+0.447i)(2,\ 175,\ (\ :9/2),\ 0.894 + 0.447i)

Particular Values

L(5)L(5) ≈\approx 2.2464521642.246452164
L(12)L(\frac12) ≈\approx 2.2464521642.246452164
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)
bad5 1 1
7 1−2.40e3iT 1 - 2.40e3iT
good2 1+14.8iT−512T2 1 + 14.8iT - 512T^{2}
3 1−239.iT−1.96e4T2 1 - 239. iT - 1.96e4T^{2}
11 1+1.65e4T+2.35e9T2 1 + 1.65e4T + 2.35e9T^{2}
13 1+2.63e4iT−1.06e10T2 1 + 2.63e4iT - 1.06e10T^{2}
17 1+1.44e5iT−1.18e11T2 1 + 1.44e5iT - 1.18e11T^{2}
19 1−1.59e5T+3.22e11T2 1 - 1.59e5T + 3.22e11T^{2}
23 1+2.07e6iT−1.80e12T2 1 + 2.07e6iT - 1.80e12T^{2}
29 1−4.94e6T+1.45e13T2 1 - 4.94e6T + 1.45e13T^{2}
31 1−4.22e6T+2.64e13T2 1 - 4.22e6T + 2.64e13T^{2}
37 1+1.29e7iT−1.29e14T2 1 + 1.29e7iT - 1.29e14T^{2}
41 1+2.87e7T+3.27e14T2 1 + 2.87e7T + 3.27e14T^{2}
43 1+3.54e7iT−5.02e14T2 1 + 3.54e7iT - 5.02e14T^{2}
47 1−5.95e7iT−1.11e15T2 1 - 5.95e7iT - 1.11e15T^{2}
53 1+2.31e6iT−3.29e15T2 1 + 2.31e6iT - 3.29e15T^{2}
59 1−1.68e8T+8.66e15T2 1 - 1.68e8T + 8.66e15T^{2}
61 1+6.70e7T+1.16e16T2 1 + 6.70e7T + 1.16e16T^{2}
67 1+1.56e8iT−2.72e16T2 1 + 1.56e8iT - 2.72e16T^{2}
71 1−6.95e7T+4.58e16T2 1 - 6.95e7T + 4.58e16T^{2}
73 1−7.83e7iT−5.88e16T2 1 - 7.83e7iT - 5.88e16T^{2}
79 1−4.26e8T+1.19e17T2 1 - 4.26e8T + 1.19e17T^{2}
83 1−5.31e8iT−1.86e17T2 1 - 5.31e8iT - 1.86e17T^{2}
89 1−1.14e8T+3.50e17T2 1 - 1.14e8T + 3.50e17T^{2}
97 1+1.46e9iT−7.60e17T2 1 + 1.46e9iT - 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

−10.68120002021342536355865471904, −10.26019652559310965512380791747, −9.316618504557064728172768295730, −8.259217413893866386339089818255, −6.62620416670885047730093130295, −5.36426541457156286212461751770, −4.34092311331886636031295134738, −3.20895386364833060935505400399, −2.41256638910707229372572248515, −0.53416562295898580710913470943, 1.03319722145540612297934964676, 1.93216098595404481962751743818, 3.09161937432092919211061403237, 5.20537749003550877479625152448, 6.33794564128637430492093249641, 6.92119247366971812386481974039, 7.82134883144711014496953838628, 8.440942829455383964489301607303, 10.19329372398641073519010588396, 11.55995211032923613848565773542

Graph of the ZZ-function along the critical line