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Properties

Label 2-650-5.4-c3-0-50
Degree 22
Conductor 650650
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 38.351238.3512
Root an. cond. 6.192836.19283
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 6.14i·3-s − 4·4-s − 12.2·6-s − 22.2i·7-s + 8i·8-s − 10.7·9-s + 43.8·11-s + 24.5i·12-s + 13i·13-s − 44.5·14-s + 16·16-s − 99.9i·17-s + 21.4i·18-s − 93.2·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.18i·3-s − 0.5·4-s − 0.835·6-s − 1.20i·7-s + 0.353i·8-s − 0.396·9-s + 1.20·11-s + 0.590i·12-s + 0.277i·13-s − 0.850·14-s + 0.250·16-s − 1.42i·17-s + 0.280i·18-s − 1.12·19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 38.351238.3512
Root analytic conductor: 6.192836.19283
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :3/2), −0.447−0.894i)(2,\ 650,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.3875070981.387507098
L(12)L(\frac12) ≈\approx 1.3875070981.387507098
L(52)L(\frac{5}{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)
bad2 1+2iT 1 + 2iT
5 1 1
13 1−13iT 1 - 13iT
good3 1+6.14iT−27T2 1 + 6.14iT - 27T^{2}
7 1+22.2iT−343T2 1 + 22.2iT - 343T^{2}
11 1−43.8T+1.33e3T2 1 - 43.8T + 1.33e3T^{2}
17 1+99.9iT−4.91e3T2 1 + 99.9iT - 4.91e3T^{2}
19 1+93.2T+6.85e3T2 1 + 93.2T + 6.85e3T^{2}
23 1+154.iT−1.21e4T2 1 + 154. iT - 1.21e4T^{2}
29 1+191.T+2.43e4T2 1 + 191.T + 2.43e4T^{2}
31 1+213.T+2.97e4T2 1 + 213.T + 2.97e4T^{2}
37 1−401.iT−5.06e4T2 1 - 401. iT - 5.06e4T^{2}
41 1−458.T+6.89e4T2 1 - 458.T + 6.89e4T^{2}
43 1+251.iT−7.95e4T2 1 + 251. iT - 7.95e4T^{2}
47 1−261.iT−1.03e5T2 1 - 261. iT - 1.03e5T^{2}
53 1−151.iT−1.48e5T2 1 - 151. iT - 1.48e5T^{2}
59 1+263.T+2.05e5T2 1 + 263.T + 2.05e5T^{2}
61 1−644.T+2.26e5T2 1 - 644.T + 2.26e5T^{2}
67 1+387.iT−3.00e5T2 1 + 387. iT - 3.00e5T^{2}
71 1+544.T+3.57e5T2 1 + 544.T + 3.57e5T^{2}
73 1−301.iT−3.89e5T2 1 - 301. iT - 3.89e5T^{2}
79 1−562.T+4.93e5T2 1 - 562.T + 4.93e5T^{2}
83 1+938.iT−5.71e5T2 1 + 938. iT - 5.71e5T^{2}
89 1+261.T+7.04e5T2 1 + 261.T + 7.04e5T^{2}
97 1−1.37e3iT−9.12e5T2 1 - 1.37e3iT - 9.12e5T^{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

−9.583134750783927182731931162000, −8.783064540855245872261815101393, −7.63564410740955374899030179025, −6.98148891071440943876842982072, −6.23450185824611668639685535895, −4.61342032717304698235071367362, −3.83871973484181012606047081883, −2.39265691802691178142350942157, −1.30887164959187708916911587928, −0.41783268480622967312699676645, 1.86005630512049669765424506836, 3.65859151428014854921268074816, 4.17164660630640255851015875600, 5.54444899720194462599125002847, 5.91560385242636328222508725378, 7.18125147808258148951357098536, 8.343281363618609285283165272451, 9.208549362300445401806837737369, 9.439144960605225633287249505673, 10.66961331563339887486717256018

Graph of the ZZ-function along the critical line