↑

Properties

Label 2-30e2-300.263-c0-0-1
Degree 22
Conductor 900900
Sign 0.414−0.910i0.414 - 0.910i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
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  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.891 − 0.453i)5-s + (−0.453 − 0.891i)8-s + (0.587 + 0.809i)10-s + (1.76 + 0.278i)13-s + (0.809 − 0.587i)16-s + (−1.04 + 0.533i)17-s + (−0.707 + 0.707i)20-s + (0.587 − 0.809i)25-s + 1.78i·26-s + (0.0966 + 0.297i)29-s + (0.707 + 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 + 1.95i)37-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.891 − 0.453i)5-s + (−0.453 − 0.891i)8-s + (0.587 + 0.809i)10-s + (1.76 + 0.278i)13-s + (0.809 − 0.587i)16-s + (−1.04 + 0.533i)17-s + (−0.707 + 0.707i)20-s + (0.587 − 0.809i)25-s + 1.78i·26-s + (0.0966 + 0.297i)29-s + (0.707 + 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 + 1.95i)37-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.414−0.910i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.414−0.910i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.414−0.910i0.414 - 0.910i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(863,⋅)\chi_{900} (863, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.414−0.910i)(2,\ 900,\ (\ :0),\ 0.414 - 0.910i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1527441681.152744168
L(12)L(\frac12) ≈\approx 1.1527441681.152744168
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+(−0.156−0.987i)T 1 + (-0.156 - 0.987i)T
3 1 1
5 1+(−0.891+0.453i)T 1 + (-0.891 + 0.453i)T
good7 1+iT2 1 + iT^{2}
11 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
13 1+(−1.76−0.278i)T+(0.951+0.309i)T2 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2}
17 1+(1.04−0.533i)T+(0.587−0.809i)T2 1 + (1.04 - 0.533i)T + (0.587 - 0.809i)T^{2}
19 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
23 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
29 1+(−0.0966−0.297i)T+(−0.809+0.587i)T2 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(0.309−1.95i)T+(−0.951−0.309i)T2 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2}
41 1+(0.533+0.734i)T+(−0.309+0.951i)T2 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
53 1+(1.69+0.863i)T+(0.587+0.809i)T2 1 + (1.69 + 0.863i)T + (0.587 + 0.809i)T^{2}
59 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
61 1+(1.53+1.11i)T+(0.309+0.951i)T2 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2}
67 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
71 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
73 1+(−0.142−0.896i)T+(−0.951+0.309i)T2 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2}
79 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
83 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
89 1+(−1.59−1.16i)T+(0.309+0.951i)T2 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2}
97 1+(0.278+0.142i)T+(0.587+0.809i)T2 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)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

−10.27806617809088493124075167601, −9.353378054353480921367976532972, −8.639364442852992271471443491798, −8.162640236360913201672480780507, −6.63612003352339336720646976819, −6.36383063511847993771795300422, −5.34819804157165677354586926108, −4.47051309866663367470360684068, −3.38825623036177170445340059149, −1.57849827473773916319013860730, 1.46904284801375981271767337577, 2.61660621865662822810301276553, 3.59926867647159677820974377101, 4.69790736560695027299841178247, 5.82195635514294145329020053793, 6.40561592206042947213489246935, 7.78619071324657404583788211471, 8.991983916961511232150284117240, 9.248312181297600262050032034655, 10.48991026934516877344822898162

Graph of the ZZ-function along the critical line