↑

Properties

Label 2-450-5.4-c3-0-12
Degree 22
Conductor 450450
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 26.550826.5508
Root an. cond. 5.152755.15275
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 − 4·4-s + 4i·7-s − 8i·8-s − 12·11-s − 58i·13-s − 8·14-s + 16·16-s + 66i·17-s + 100·19-s − 24i·22-s − 132i·23-s + 116·26-s − 16i·28-s − 90·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.215i·7-s − 0.353i·8-s − 0.328·11-s − 1.23i·13-s − 0.152·14-s + 0.250·16-s + 0.941i·17-s + 1.20·19-s − 0.232i·22-s − 1.19i·23-s + 0.874·26-s − 0.107i·28-s − 0.576·29-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{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: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 26.550826.5508
Root analytic conductor: 5.152755.15275
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ450(199,⋅)\chi_{450} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 450, ( :3/2), 0.894−0.447i)(2,\ 450,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.7185246621.718524662
L(12)L(\frac12) ≈\approx 1.7185246621.718524662
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
3 1 1
5 1 1
good7 1−4iT−343T2 1 - 4iT - 343T^{2}
11 1+12T+1.33e3T2 1 + 12T + 1.33e3T^{2}
13 1+58iT−2.19e3T2 1 + 58iT - 2.19e3T^{2}
17 1−66iT−4.91e3T2 1 - 66iT - 4.91e3T^{2}
19 1−100T+6.85e3T2 1 - 100T + 6.85e3T^{2}
23 1+132iT−1.21e4T2 1 + 132iT - 1.21e4T^{2}
29 1+90T+2.43e4T2 1 + 90T + 2.43e4T^{2}
31 1−152T+2.97e4T2 1 - 152T + 2.97e4T^{2}
37 1−34iT−5.06e4T2 1 - 34iT - 5.06e4T^{2}
41 1−438T+6.89e4T2 1 - 438T + 6.89e4T^{2}
43 1−32iT−7.95e4T2 1 - 32iT - 7.95e4T^{2}
47 1+204iT−1.03e5T2 1 + 204iT - 1.03e5T^{2}
53 1+222iT−1.48e5T2 1 + 222iT - 1.48e5T^{2}
59 1−420T+2.05e5T2 1 - 420T + 2.05e5T^{2}
61 1−902T+2.26e5T2 1 - 902T + 2.26e5T^{2}
67 1−1.02e3iT−3.00e5T2 1 - 1.02e3iT - 3.00e5T^{2}
71 1+432T+3.57e5T2 1 + 432T + 3.57e5T^{2}
73 1−362iT−3.89e5T2 1 - 362iT - 3.89e5T^{2}
79 1−160T+4.93e5T2 1 - 160T + 4.93e5T^{2}
83 1+72iT−5.71e5T2 1 + 72iT - 5.71e5T^{2}
89 1−810T+7.04e5T2 1 - 810T + 7.04e5T^{2}
97 1+1.10e3iT−9.12e5T2 1 + 1.10e3iT - 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

−10.51804494624323660398869690701, −9.867074630777899289913721191661, −8.671545843867980458318146727698, −8.019383443235623041770190810278, −7.09416946760370309077869307043, −5.93781691166390590797326251258, −5.26132311540951957578189144581, −3.98117166468712184266240546502, −2.67582355966347299025569646735, −0.75602300209059065478118762817, 0.959846312424796667948307735738, 2.34921737102649215019836779592, 3.56924481079303792323852415316, 4.65756054180351006520017865160, 5.68093892312792666205441509627, 7.05372382141228285960933022142, 7.85307847814056745705920043101, 9.241127417908519745188081306337, 9.560657420168125583345953404815, 10.72848523748022034471888247187

Graph of the ZZ-function along the critical line