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Properties

Label 2-3192-3192.797-c0-0-20
Degree 22
Conductor 31923192
Sign 11
Analytic cond. 1.593011.59301
Root an. cond. 1.262141.26214
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  + 2-s − i·3-s + 4-s + 2i·5-s − i·6-s + 7-s + 8-s − 9-s + 2i·10-s − i·12-s − 2i·13-s + 14-s + 2·15-s + 16-s − 18-s + i·19-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s + 2i·5-s − i·6-s + 7-s + 8-s − 9-s + 2i·10-s − i·12-s − 2i·13-s + 14-s + 2·15-s + 16-s − 18-s + i·19-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3192(797,⋅)\chi_{3192} (797, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3192, ( :0), 1)(2,\ 3192,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.6269034332.626903433
L(12)L(\frac12) ≈\approx 2.6269034332.626903433
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−T 1 - T
3 1+iT 1 + iT
7 1−T 1 - T
19 1−iT 1 - iT
good5 1−2iT−T2 1 - 2iT - T^{2}
11 1+T2 1 + T^{2}
13 1+2iT−T2 1 + 2iT - T^{2}
17 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1−T2 1 - T^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 1+T2 1 + T^{2}
71 1+2T+T2 1 + 2T + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−2iT−T2 1 - 2iT - T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + T^{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

−8.268615859376721238835279066319, −7.77890167391301296388048214636, −7.33282346279644752314708031636, −6.55370510825433869526027986228, −5.80996117496152935740645317042, −5.39344144220987535587836465336, −3.93919233747926946379096691669, −3.06905352953739779879424344563, −2.58487375487441571902150229378, −1.57876075143363064038193750824, 1.40277724394777344256169987574, 2.29589231477238962021497273116, 3.82072583867059722238170886956, 4.50017289460094680775993294829, 4.72465966697653627683405540611, 5.40744082925062587124427262978, 6.24256198640426821314259863087, 7.38131304902125344300158087262, 8.264928951684689707314588135520, 8.967346389423622737468225852640

Graph of the ZZ-function along the critical line