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Properties

Label 2-525-5.4-c3-0-27
Degree 22
Conductor 525525
Sign 0.447−0.894i0.447 - 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  + 4.53i·2-s − 3i·3-s − 12.5·4-s + 13.5·6-s − 7i·7-s − 20.5i·8-s − 9·9-s − 19.0·11-s + 37.5i·12-s + 2.93i·13-s + 31.7·14-s − 7.21·16-s − 6.49i·17-s − 40.7i·18-s + 5.43·19-s + ⋯
L(s)  = 1  + 1.60i·2-s − 0.577i·3-s − 1.56·4-s + 0.924·6-s − 0.377i·7-s − 0.907i·8-s − 0.333·9-s − 0.522·11-s + 0.904i·12-s + 0.0626i·13-s + 0.605·14-s − 0.112·16-s − 0.0927i·17-s − 0.533i·18-s + 0.0656·19-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: 0.447−0.894i0.447 - 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 1.5958283171.595828317
L(12)L(\frac12) ≈\approx 1.5958283171.595828317
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 1−4.53iT−8T2 1 - 4.53iT - 8T^{2}
11 1+19.0T+1.33e3T2 1 + 19.0T + 1.33e3T^{2}
13 1−2.93iT−2.19e3T2 1 - 2.93iT - 2.19e3T^{2}
17 1+6.49iT−4.91e3T2 1 + 6.49iT - 4.91e3T^{2}
19 1−5.43T+6.85e3T2 1 - 5.43T + 6.85e3T^{2}
23 1+49.3iT−1.21e4T2 1 + 49.3iT - 1.21e4T^{2}
29 1−291.T+2.43e4T2 1 - 291.T + 2.43e4T^{2}
31 1−244.T+2.97e4T2 1 - 244.T + 2.97e4T^{2}
37 1+193.iT−5.06e4T2 1 + 193. iT - 5.06e4T^{2}
41 1−315.T+6.89e4T2 1 - 315.T + 6.89e4T^{2}
43 1−300.iT−7.95e4T2 1 - 300. iT - 7.95e4T^{2}
47 1−86.5iT−1.03e5T2 1 - 86.5iT - 1.03e5T^{2}
53 1+509.iT−1.48e5T2 1 + 509. iT - 1.48e5T^{2}
59 1−83.3T+2.05e5T2 1 - 83.3T + 2.05e5T^{2}
61 1+5.25T+2.26e5T2 1 + 5.25T + 2.26e5T^{2}
67 1−205.iT−3.00e5T2 1 - 205. iT - 3.00e5T^{2}
71 1−1.00e3T+3.57e5T2 1 - 1.00e3T + 3.57e5T^{2}
73 1−1.00e3iT−3.89e5T2 1 - 1.00e3iT - 3.89e5T^{2}
79 1−863.T+4.93e5T2 1 - 863.T + 4.93e5T^{2}
83 1+1.33e3iT−5.71e5T2 1 + 1.33e3iT - 5.71e5T^{2}
89 1+326.T+7.04e5T2 1 + 326.T + 7.04e5T^{2}
97 1−1.52e3iT−9.12e5T2 1 - 1.52e3iT - 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

−10.45329260520514514942915328639, −9.401933122833128782539599914653, −8.301017886816017981260461214780, −7.891179616727495944599954178332, −6.85758866460490217686195210065, −6.31714700248074349796356714708, −5.24789212667538642918090425330, −4.33430164799673217844166853276, −2.62292098925399939831972819075, −0.69665451496178968429292956999, 0.868343270078126560103328904762, 2.37717022641922166950124653938, 3.16997452289445418459853478361, 4.31358716731988401315353683963, 5.15095990825133038978404529576, 6.45658335329951133032491410051, 8.016587974664946570600668043427, 8.887331437523068612966163298906, 9.757080419167479103519571713321, 10.37886639460685881227761049474

Graph of the ZZ-function along the critical line