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Properties

Label 2-525-5.4-c3-0-23
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  + 1.70i·2-s + 3i·3-s + 5.10·4-s − 5.10·6-s − 7i·7-s + 22.2i·8-s − 9·9-s + 37.4·11-s + 15.3i·12-s + 29.0i·13-s + 11.9·14-s + 2.89·16-s − 58.4i·17-s − 15.3i·18-s + 54.5·19-s + ⋯
L(s)  = 1  + 0.601i·2-s + 0.577i·3-s + 0.638·4-s − 0.347·6-s − 0.377i·7-s + 0.985i·8-s − 0.333·9-s + 1.02·11-s + 0.368i·12-s + 0.619i·13-s + 0.227·14-s + 0.0452·16-s − 0.833i·17-s − 0.200i·18-s + 0.659·19-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 2.4831733842.483173384
L(12)L(\frac12) ≈\approx 2.4831733842.483173384
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 1−1.70iT−8T2 1 - 1.70iT - 8T^{2}
11 1−37.4T+1.33e3T2 1 - 37.4T + 1.33e3T^{2}
13 1−29.0iT−2.19e3T2 1 - 29.0iT - 2.19e3T^{2}
17 1+58.4iT−4.91e3T2 1 + 58.4iT - 4.91e3T^{2}
19 1−54.5T+6.85e3T2 1 - 54.5T + 6.85e3T^{2}
23 1−161.iT−1.21e4T2 1 - 161. iT - 1.21e4T^{2}
29 1+137.T+2.43e4T2 1 + 137.T + 2.43e4T^{2}
31 1−154.T+2.97e4T2 1 - 154.T + 2.97e4T^{2}
37 1−350.iT−5.06e4T2 1 - 350. iT - 5.06e4T^{2}
41 1−353.T+6.89e4T2 1 - 353.T + 6.89e4T^{2}
43 1+518.iT−7.95e4T2 1 + 518. iT - 7.95e4T^{2}
47 1−542.iT−1.03e5T2 1 - 542. iT - 1.03e5T^{2}
53 1−305.iT−1.48e5T2 1 - 305. iT - 1.48e5T^{2}
59 1+14.6T+2.05e5T2 1 + 14.6T + 2.05e5T^{2}
61 1+171.T+2.26e5T2 1 + 171.T + 2.26e5T^{2}
67 1+551.iT−3.00e5T2 1 + 551. iT - 3.00e5T^{2}
71 1+120.T+3.57e5T2 1 + 120.T + 3.57e5T^{2}
73 1−284.iT−3.89e5T2 1 - 284. iT - 3.89e5T^{2}
79 1+941.T+4.93e5T2 1 + 941.T + 4.93e5T^{2}
83 1−377.iT−5.71e5T2 1 - 377. iT - 5.71e5T^{2}
89 1−677.T+7.04e5T2 1 - 677.T + 7.04e5T^{2}
97 1−1.22e3iT−9.12e5T2 1 - 1.22e3iT - 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

−10.83680943085414779157017277532, −9.672263532239419960032676827589, −9.088882900762429916558496784427, −7.83092551379854683403825392977, −7.10623925716719638895432003291, −6.21225056256950016229078713362, −5.24277622308417801666530011906, −4.10584405886855613180340072901, −2.93953450709289900493870022123, −1.40047953370695651041519233341, 0.798245066365011784058325706290, 1.94162757884440659143311000644, 2.98568718307386687278641873315, 4.12312531176882143638939626380, 5.75731937302738976463607514912, 6.46760973555944416210231163172, 7.38774583064520151755923677994, 8.369270503816395697422366970009, 9.379700224307719900981578086573, 10.34077258387896157738125198695

Graph of the ZZ-function along the critical line