↑

Properties

Label 2-450-5.4-c3-0-3
Degree 22
Conductor 450450
Sign 0.447−0.894i0.447 - 0.894i
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 + i·7-s + 8i·8-s − 42·11-s − 67i·13-s + 2·14-s + 16·16-s + 54i·17-s + 115·19-s + 84i·22-s + 162i·23-s − 134·26-s − 4i·28-s − 210·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.0539i·7-s + 0.353i·8-s − 1.15·11-s − 1.42i·13-s + 0.0381·14-s + 0.250·16-s + 0.770i·17-s + 1.38·19-s + 0.814i·22-s + 1.46i·23-s − 1.01·26-s − 0.0269i·28-s − 1.34·29-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{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: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
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), 0.447−0.894i)(2,\ 450,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.83256639090.8325663909
L(12)L(\frac12) ≈\approx 0.83256639090.8325663909
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−iT−343T2 1 - iT - 343T^{2}
11 1+42T+1.33e3T2 1 + 42T + 1.33e3T^{2}
13 1+67iT−2.19e3T2 1 + 67iT - 2.19e3T^{2}
17 1−54iT−4.91e3T2 1 - 54iT - 4.91e3T^{2}
19 1−115T+6.85e3T2 1 - 115T + 6.85e3T^{2}
23 1−162iT−1.21e4T2 1 - 162iT - 1.21e4T^{2}
29 1+210T+2.43e4T2 1 + 210T + 2.43e4T^{2}
31 1+193T+2.97e4T2 1 + 193T + 2.97e4T^{2}
37 1−286iT−5.06e4T2 1 - 286iT - 5.06e4T^{2}
41 1+12T+6.89e4T2 1 + 12T + 6.89e4T^{2}
43 1−263iT−7.95e4T2 1 - 263iT - 7.95e4T^{2}
47 1−414iT−1.03e5T2 1 - 414iT - 1.03e5T^{2}
53 1−192iT−1.48e5T2 1 - 192iT - 1.48e5T^{2}
59 1−690T+2.05e5T2 1 - 690T + 2.05e5T^{2}
61 1+733T+2.26e5T2 1 + 733T + 2.26e5T^{2}
67 1+299iT−3.00e5T2 1 + 299iT - 3.00e5T^{2}
71 1−228T+3.57e5T2 1 - 228T + 3.57e5T^{2}
73 1−938iT−3.89e5T2 1 - 938iT - 3.89e5T^{2}
79 1−160T+4.93e5T2 1 - 160T + 4.93e5T^{2}
83 1−462iT−5.71e5T2 1 - 462iT - 5.71e5T^{2}
89 1+240T+7.04e5T2 1 + 240T + 7.04e5T^{2}
97 1−511iT−9.12e5T2 1 - 511iT - 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

−10.83873681475319718413332433348, −10.04631242273619391487746896641, −9.276509670919285682082372819313, −8.004239565543575578434790469912, −7.52123912447999566358368013415, −5.72425587523346107784076344787, −5.20584491168773341484575028387, −3.65225842048490600129525126996, −2.78991282878349325069165891270, −1.28978331279074381337272596413, 0.28117832935911331330552939217, 2.20996701283792806053974829948, 3.73581182558786197460205492013, 4.93138856214640589143034796965, 5.71360486807984569434128470588, 7.03886836608338327275433017618, 7.48998783759318748165935589343, 8.739591927397558823403244219767, 9.402775764559550739188966706582, 10.42358364976021129169946794179

Graph of the ZZ-function along the critical line