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Properties

Label 2-2e5-4.3-c16-0-4
Degree 22
Conductor 3232
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 51.943851.9438
Root an. cond. 7.207207.20720
Motivic weight 1616
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.24e4i·3-s − 5.59e5·5-s + 8.71e6i·7-s − 1.11e8·9-s − 1.47e8i·11-s + 4.14e8·13-s + 6.95e9i·15-s − 6.31e8·17-s + 2.19e10i·19-s + 1.08e11·21-s − 7.67e10i·23-s + 1.60e11·25-s + 8.52e11i·27-s + 1.23e11·29-s − 1.90e11i·31-s + ⋯
L(s)  = 1  − 1.89i·3-s − 1.43·5-s + 1.51i·7-s − 2.59·9-s − 0.686i·11-s + 0.508·13-s + 2.71i·15-s − 0.0905·17-s + 1.29i·19-s + 2.86·21-s − 0.979i·23-s + 1.04·25-s + 3.01i·27-s + 0.246·29-s − 0.223i·31-s + ⋯

Functional equation

Λ(s)=(32s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(17−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(17-s) \end{aligned}
Λ(s)=(32s/2 ΓC(s+8) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3232    =    252^{5}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 51.943851.9438
Root analytic conductor: 7.207207.20720
Motivic weight: 1616
Rational: no
Arithmetic: yes
Character: χ32(31,⋅)\chi_{32} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 32, ( :8), 0.707+0.707i)(2,\ 32,\ (\ :8),\ 0.707 + 0.707i)

Particular Values

L(172)L(\frac{17}{2}) ≈\approx 1.0075471621.007547162
L(12)L(\frac12) ≈\approx 1.0075471621.007547162
L(9)L(9) 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
good3 1+1.24e4iT−4.30e7T2 1 + 1.24e4iT - 4.30e7T^{2}
5 1+5.59e5T+1.52e11T2 1 + 5.59e5T + 1.52e11T^{2}
7 1−8.71e6iT−3.32e13T2 1 - 8.71e6iT - 3.32e13T^{2}
11 1+1.47e8iT−4.59e16T2 1 + 1.47e8iT - 4.59e16T^{2}
13 1−4.14e8T+6.65e17T2 1 - 4.14e8T + 6.65e17T^{2}
17 1+6.31e8T+4.86e19T2 1 + 6.31e8T + 4.86e19T^{2}
19 1−2.19e10iT−2.88e20T2 1 - 2.19e10iT - 2.88e20T^{2}
23 1+7.67e10iT−6.13e21T2 1 + 7.67e10iT - 6.13e21T^{2}
29 1−1.23e11T+2.50e23T2 1 - 1.23e11T + 2.50e23T^{2}
31 1+1.90e11iT−7.27e23T2 1 + 1.90e11iT - 7.27e23T^{2}
37 1+5.54e11T+1.23e25T2 1 + 5.54e11T + 1.23e25T^{2}
41 1+1.32e13T+6.37e25T2 1 + 1.32e13T + 6.37e25T^{2}
43 1+4.26e12iT−1.36e26T2 1 + 4.26e12iT - 1.36e26T^{2}
47 1−8.72e12iT−5.66e26T2 1 - 8.72e12iT - 5.66e26T^{2}
53 1−3.66e13T+3.87e27T2 1 - 3.66e13T + 3.87e27T^{2}
59 1+1.78e14iT−2.15e28T2 1 + 1.78e14iT - 2.15e28T^{2}
61 1−2.89e14T+3.67e28T2 1 - 2.89e14T + 3.67e28T^{2}
67 1−1.70e14iT−1.64e29T2 1 - 1.70e14iT - 1.64e29T^{2}
71 1+2.91e14iT−4.16e29T2 1 + 2.91e14iT - 4.16e29T^{2}
73 1−1.24e15T+6.50e29T2 1 - 1.24e15T + 6.50e29T^{2}
79 1−1.72e15iT−2.30e30T2 1 - 1.72e15iT - 2.30e30T^{2}
83 1−4.17e14iT−5.07e30T2 1 - 4.17e14iT - 5.07e30T^{2}
89 1+3.35e15T+1.54e31T2 1 + 3.35e15T + 1.54e31T^{2}
97 1+6.85e15T+6.14e31T2 1 + 6.85e15T + 6.14e31T^{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

−12.70800504334046819462381901778, −12.05486290711513256699518249201, −11.32313843439902066358274588193, −8.473802203284402731922901211792, −8.195591618294503718819373207463, −6.72862620168617901389096744341, −5.64531979636092292679654861362, −3.32533721233090034429773653616, −2.05747153682340354106658558534, −0.64302509466326376259400254133, 0.46502023751953663884970447002, 3.35869895965424607285416877588, 4.07752096843611863361843207020, 4.89089370096687217084201458506, 7.15338584363001012396915967469, 8.516347665602809376446659698438, 9.883588361475971481214946493752, 10.85961155799588195300359372728, 11.61962648183551165133214368404, 13.66426004629681264731906362692

Graph of the ZZ-function along the critical line