↑

Properties

Label 2-525-5.4-c3-0-52
Degree 22
Conductor 525525
Sign −0.894−0.447i-0.894 - 0.447i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 3i·3-s + 4·4-s − 6·6-s − 7i·7-s − 24i·8-s − 9·9-s − 21·11-s − 12i·12-s − 24i·13-s − 14·14-s − 16·16-s − 22i·17-s + 18i·18-s − 16·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s + 0.5·4-s − 0.408·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s − 0.575·11-s − 0.288i·12-s − 0.512i·13-s − 0.267·14-s − 0.250·16-s − 0.313i·17-s + 0.235i·18-s − 0.193·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.894−0.447i-0.894 - 0.447i
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), −0.894−0.447i)(2,\ 525,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.3823613151.382361315
L(12)L(\frac12) ≈\approx 1.3823613151.382361315
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+2iT−8T2 1 + 2iT - 8T^{2}
11 1+21T+1.33e3T2 1 + 21T + 1.33e3T^{2}
13 1+24iT−2.19e3T2 1 + 24iT - 2.19e3T^{2}
17 1+22iT−4.91e3T2 1 + 22iT - 4.91e3T^{2}
19 1+16T+6.85e3T2 1 + 16T + 6.85e3T^{2}
23 1−25iT−1.21e4T2 1 - 25iT - 1.21e4T^{2}
29 1+167T+2.43e4T2 1 + 167T + 2.43e4T^{2}
31 1−10T+2.97e4T2 1 - 10T + 2.97e4T^{2}
37 1+133iT−5.06e4T2 1 + 133iT - 5.06e4T^{2}
41 1+168T+6.89e4T2 1 + 168T + 6.89e4T^{2}
43 1−97iT−7.95e4T2 1 - 97iT - 7.95e4T^{2}
47 1+400iT−1.03e5T2 1 + 400iT - 1.03e5T^{2}
53 1−182iT−1.48e5T2 1 - 182iT - 1.48e5T^{2}
59 1+488T+2.05e5T2 1 + 488T + 2.05e5T^{2}
61 1−28T+2.26e5T2 1 - 28T + 2.26e5T^{2}
67 1+967iT−3.00e5T2 1 + 967iT - 3.00e5T^{2}
71 1+285T+3.57e5T2 1 + 285T + 3.57e5T^{2}
73 1−838iT−3.89e5T2 1 - 838iT - 3.89e5T^{2}
79 1−469T+4.93e5T2 1 - 469T + 4.93e5T^{2}
83 1−406iT−5.71e5T2 1 - 406iT - 5.71e5T^{2}
89 1+324T+7.04e5T2 1 + 324T + 7.04e5T^{2}
97 1+114iT−9.12e5T2 1 + 114iT - 9.12e5T^{2}
show more
show less
   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

−10.21420790210294167924953949616, −9.223980873564047010625797149322, −7.948282652397653849486575767798, −7.28808761962150163831461415376, −6.35826795053767554196357805109, −5.26994691689554823239706844716, −3.77574623727867459499807250738, −2.75288179062054437215414622114, −1.68071640212912531972121827183, −0.38142142708802669271933037256, 1.94354790731588735069840882354, 3.11999090111784363946957797728, 4.54174436188252590321305161765, 5.53096164188063286123335215275, 6.30861724904856560383403885985, 7.33274278909123766119537041455, 8.228796414578987652865381425721, 9.034035468909475440393734547129, 10.09822296234234106787534183967, 10.95434016999755428996981208780

Graph of the ZZ-function along the critical line