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Properties

Label 2-15e3-3.2-c0-0-1
Degree 22
Conductor 33753375
Sign −1-1
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
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  + 1.33i·2-s − 0.790·4-s + 0.279i·8-s − 1.16·16-s + 1.82i·17-s − 1.33·19-s + 0.209i·23-s + 1.82·31-s − 1.27i·32-s − 2.44·34-s − 1.79i·38-s − 0.279·46-s + 0.618i·47-s − 49-s + 1.95i·53-s + ⋯
L(s)  = 1  + 1.33i·2-s − 0.790·4-s + 0.279i·8-s − 1.16·16-s + 1.82i·17-s − 1.33·19-s + 0.209i·23-s + 1.82·31-s − 1.27i·32-s − 2.44·34-s − 1.79i·38-s − 0.279·46-s + 0.618i·47-s − 49-s + 1.95i·53-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(1376,⋅)\chi_{3375} (1376, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3375, ( :0), −1)(2,\ 3375,\ (\ :0),\ -1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0494469221.049446922
L(12)L(\frac12) ≈\approx 1.0494469221.049446922
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)
bad3 1 1
5 1 1
good2 1−1.33iT−T2 1 - 1.33iT - T^{2}
7 1+T2 1 + T^{2}
11 1−T2 1 - T^{2}
13 1+T2 1 + T^{2}
17 1−1.82iT−T2 1 - 1.82iT - T^{2}
19 1+1.33T+T2 1 + 1.33T + T^{2}
23 1−0.209iT−T2 1 - 0.209iT - T^{2}
29 1−T2 1 - T^{2}
31 1−1.82T+T2 1 - 1.82T + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−0.618iT−T2 1 - 0.618iT - T^{2}
53 1−1.95iT−T2 1 - 1.95iT - T^{2}
59 1−T2 1 - T^{2}
61 1+1.95T+T2 1 + 1.95T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−1.95T+T2 1 - 1.95T + T^{2}
83 1−1.95iT−T2 1 - 1.95iT - 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

−8.825222737480490100230818227444, −8.154570927242880399863047237480, −7.83745466537385128034347356573, −6.71546551106570761506650132229, −6.29259195118030257700162013294, −5.72388213842788098851298998120, −4.65317491767754539981333880651, −4.10556150031445727635030103659, −2.79407767424672084932317844664, −1.66656515019787610549979213440, 0.60509716381761286833321176178, 1.91836205548493902795633301186, 2.69867211466158488887812484968, 3.43224047465848575120971854693, 4.48122096768270992876452055487, 4.96673405790260399561103164244, 6.31324908031131142853839407029, 6.82577775678443202043494634617, 7.81729282935822541065844827956, 8.666534867949707468184752699091

Graph of the ZZ-function along the critical line