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Properties

Label 2-600-5.3-c2-0-8
Degree 22
Conductor 600600
Sign 0.608−0.793i0.608 - 0.793i
Analytic cond. 16.348816.3488
Root an. cond. 4.043364.04336
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (0.775 − 0.775i)7-s + 2.99i·9-s + 2.89·11-s + (5.87 + 5.87i)13-s + (4.44 − 4.44i)17-s − 0.101i·19-s + 1.89·21-s + (25.3 + 25.3i)23-s + (−3.67 + 3.67i)27-s + 32.2i·29-s − 3.69·31-s + (3.55 + 3.55i)33-s + (42.6 − 42.6i)37-s + 14.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.110 − 0.110i)7-s + 0.333i·9-s + 0.263·11-s + (0.452 + 0.452i)13-s + (0.261 − 0.261i)17-s − 0.00531i·19-s + 0.0904·21-s + (1.10 + 1.10i)23-s + (−0.136 + 0.136i)27-s + 1.11i·29-s − 0.119·31-s + (0.107 + 0.107i)33-s + (1.15 − 1.15i)37-s + 0.369i·39-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=((0.608−0.793i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+1) L(s)=((0.608−0.793i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 0.608−0.793i0.608 - 0.793i
Analytic conductor: 16.348816.3488
Root analytic conductor: 4.043364.04336
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ600(193,⋅)\chi_{600} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 600, ( :1), 0.608−0.793i)(2,\ 600,\ (\ :1),\ 0.608 - 0.793i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.1539002522.153900252
L(12)L(\frac12) ≈\approx 2.1539002522.153900252
L(2)L(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 1
3 1+(−1.22−1.22i)T 1 + (-1.22 - 1.22i)T
5 1 1
good7 1+(−0.775+0.775i)T−49iT2 1 + (-0.775 + 0.775i)T - 49iT^{2}
11 1−2.89T+121T2 1 - 2.89T + 121T^{2}
13 1+(−5.87−5.87i)T+169iT2 1 + (-5.87 - 5.87i)T + 169iT^{2}
17 1+(−4.44+4.44i)T−289iT2 1 + (-4.44 + 4.44i)T - 289iT^{2}
19 1+0.101iT−361T2 1 + 0.101iT - 361T^{2}
23 1+(−25.3−25.3i)T+529iT2 1 + (-25.3 - 25.3i)T + 529iT^{2}
29 1−32.2iT−841T2 1 - 32.2iT - 841T^{2}
31 1+3.69T+961T2 1 + 3.69T + 961T^{2}
37 1+(−42.6+42.6i)T−1.36e3iT2 1 + (-42.6 + 42.6i)T - 1.36e3iT^{2}
41 1+12.8T+1.68e3T2 1 + 12.8T + 1.68e3T^{2}
43 1+(−49.2−49.2i)T+1.84e3iT2 1 + (-49.2 - 49.2i)T + 1.84e3iT^{2}
47 1+(−2.85+2.85i)T−2.20e3iT2 1 + (-2.85 + 2.85i)T - 2.20e3iT^{2}
53 1+(13.1+13.1i)T+2.80e3iT2 1 + (13.1 + 13.1i)T + 2.80e3iT^{2}
59 1−76.3iT−3.48e3T2 1 - 76.3iT - 3.48e3T^{2}
61 1+103.T+3.72e3T2 1 + 103.T + 3.72e3T^{2}
67 1+(−47.6+47.6i)T−4.48e3iT2 1 + (-47.6 + 47.6i)T - 4.48e3iT^{2}
71 1−29.7T+5.04e3T2 1 - 29.7T + 5.04e3T^{2}
73 1+(−3.50−3.50i)T+5.32e3iT2 1 + (-3.50 - 3.50i)T + 5.32e3iT^{2}
79 1+87.7iT−6.24e3T2 1 + 87.7iT - 6.24e3T^{2}
83 1+(−81.7−81.7i)T+6.88e3iT2 1 + (-81.7 - 81.7i)T + 6.88e3iT^{2}
89 1−96.5iT−7.92e3T2 1 - 96.5iT - 7.92e3T^{2}
97 1+(−54.2+54.2i)T−9.40e3iT2 1 + (-54.2 + 54.2i)T - 9.40e3iT^{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

−10.72859302123912347981627690782, −9.386271485676022805314637455659, −9.158920161841534376789161208601, −7.934405188335807505961091961328, −7.15608564776996722058021381486, −5.99707032142572137861563293168, −4.92316549027702990729327825434, −3.90695098922698027938242351174, −2.87581195288427901548100638631, −1.33936140956052270359159840803, 0.892343155790627957989094404399, 2.35242428200595951349936936704, 3.48676679721458799471014036754, 4.67721602825181243574222086979, 5.91316601444185776173671017552, 6.74059370702176364455561503320, 7.79816672452001124238935212130, 8.501334482097550432416695330238, 9.348397151508070995894079109513, 10.33263660974649146035057332871

Graph of the ZZ-function along the critical line