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Properties

Label 2-525-5.4-c3-0-49
Degree 22
Conductor 525525
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 30.976030.9760
Root an. cond. 5.565605.56560
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  − 4.70i·2-s + 3i·3-s − 14.1·4-s + 14.1·6-s − 7i·7-s + 28.7i·8-s − 9·9-s + 24.5·11-s − 42.3i·12-s − 35.0i·13-s − 32.9·14-s + 22.1·16-s + 18.4i·17-s + 42.3i·18-s + 67.4·19-s + ⋯
L(s)  = 1  − 1.66i·2-s + 0.577i·3-s − 1.76·4-s + 0.959·6-s − 0.377i·7-s + 1.26i·8-s − 0.333·9-s + 0.674·11-s − 1.01i·12-s − 0.747i·13-s − 0.628·14-s + 0.345·16-s + 0.262i·17-s + 0.554i·18-s + 0.813·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{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: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 30.976030.9760
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ525(274,⋅)\chi_{525} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :3/2), −0.447−0.894i)(2,\ 525,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.71145518500.7114551850
L(12)L(\frac12) ≈\approx 0.71145518500.7114551850
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)
bad3 1−3iT 1 - 3iT
5 1 1
7 1+7iT 1 + 7iT
good2 1+4.70iT−8T2 1 + 4.70iT - 8T^{2}
11 1−24.5T+1.33e3T2 1 - 24.5T + 1.33e3T^{2}
13 1+35.0iT−2.19e3T2 1 + 35.0iT - 2.19e3T^{2}
17 1−18.4iT−4.91e3T2 1 - 18.4iT - 4.91e3T^{2}
19 1−67.4T+6.85e3T2 1 - 67.4T + 6.85e3T^{2}
23 1+145.iT−1.21e4T2 1 + 145. iT - 1.21e4T^{2}
29 1+214.T+2.43e4T2 1 + 214.T + 2.43e4T^{2}
31 1+88.6T+2.97e4T2 1 + 88.6T + 2.97e4T^{2}
37 1+162.iT−5.06e4T2 1 + 162. iT - 5.06e4T^{2}
41 1+337.T+6.89e4T2 1 + 337.T + 6.89e4T^{2}
43 1−122.iT−7.95e4T2 1 - 122. iT - 7.95e4T^{2}
47 1+354.iT−1.03e5T2 1 + 354. iT - 1.03e5T^{2}
53 1−676.iT−1.48e5T2 1 - 676. iT - 1.48e5T^{2}
59 1+501.T+2.05e5T2 1 + 501.T + 2.05e5T^{2}
61 1+708.T+2.26e5T2 1 + 708.T + 2.26e5T^{2}
67 1−907.iT−3.00e5T2 1 - 907. iT - 3.00e5T^{2}
71 1−430.T+3.57e5T2 1 - 430.T + 3.57e5T^{2}
73 1−41.3iT−3.89e5T2 1 - 41.3iT - 3.89e5T^{2}
79 1+890.T+4.93e5T2 1 + 890.T + 4.93e5T^{2}
83 1+1.05e3iT−5.71e5T2 1 + 1.05e3iT - 5.71e5T^{2}
89 1+1.47e3T+7.04e5T2 1 + 1.47e3T + 7.04e5T^{2}
97 1+555.iT−9.12e5T2 1 + 555. iT - 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

−10.11742334010186195950749351207, −9.309617137185065630889621205172, −8.557009241930014478857564783659, −7.25097390259322107168228324747, −5.78314586977567984563206079364, −4.60491514471741011187902883542, −3.76250317392425434015619871047, −2.90805334431274191067107390342, −1.53153012872217405322416020043, −0.22118455616605298102314122836, 1.66175388036912982330645776556, 3.54077301345161542454763528334, 4.91664470153598504592470692001, 5.74711471816309571751595919773, 6.61047134681383072474328247354, 7.32096340095898246774710912719, 8.060201619353959982714263624243, 9.124070215834131592812891800468, 9.529287500260280904841921732733, 11.27739763172448817903954395878

Graph of the ZZ-function along the critical line