↑

Properties

Label 2-60e2-16.11-c0-0-0
Degree 22
Conductor 36003600
Sign −0.382+0.923i-0.382 + 0.923i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 − i)19-s + 1.41·23-s − 2i·31-s + (−0.707 + 0.707i)32-s + (1.00 − 1.00i)34-s − 1.41·38-s + (1.00 − 1.00i)46-s − 1.41i·47-s − 49-s + (−1 + i)61-s + (−1.41 − 1.41i)62-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 − i)19-s + 1.41·23-s − 2i·31-s + (−0.707 + 0.707i)32-s + (1.00 − 1.00i)34-s − 1.41·38-s + (1.00 − 1.00i)46-s − 1.41i·47-s − 49-s + (−1 + i)61-s + (−1.41 − 1.41i)62-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: −0.382+0.923i-0.382 + 0.923i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(2251,⋅)\chi_{3600} (2251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), −0.382+0.923i)(2,\ 3600,\ (\ :0),\ -0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7887827111.788782711
L(12)L(\frac12) ≈\approx 1.7887827111.788782711
L(1)L(1) 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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
3 1 1
5 1 1
good7 1+T2 1 + T^{2}
11 1−iT2 1 - iT^{2}
13 1−iT2 1 - iT^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
23 1−1.41T+T2 1 - 1.41T + T^{2}
29 1−iT2 1 - iT^{2}
31 1+2iT−T2 1 + 2iT - T^{2}
37 1+iT2 1 + iT^{2}
41 1−T2 1 - T^{2}
43 1−iT2 1 - iT^{2}
47 1+1.41iT−T2 1 + 1.41iT - T^{2}
53 1+iT2 1 + iT^{2}
59 1−iT2 1 - iT^{2}
61 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + T^{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

−8.677544458126776470747749793296, −7.69219125961290442065247765548, −6.86213226580840202940040756303, −6.10727422809373159310536179858, −5.33182694746686031333768168516, −4.64682768377721411346024987088, −3.78500489946826328544568450396, −2.96059460854360747427595305849, −2.11342649451809806666003523781, −0.864490264831248719013907376912, 1.56873612633801131118794968377, 2.98954593424674399346939061585, 3.49023799246816021736920406591, 4.57505655808519907944404620231, 5.17550548795675326349954246620, 6.00477529917657942880727165907, 6.63365072542530337417233310140, 7.44391627976619309053602629073, 8.064195037630132165942152590804, 8.739548244820448964306553226188

Graph of the ZZ-function along the critical line