↑

Properties

Label 2-105e2-1.1-c1-0-35
Degree 22
Conductor 1102511025
Sign −1-1
Analytic cond. 88.035088.0350
Root an. cond. 9.382709.38270
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 6·13-s − 16-s − 2·17-s + 8·19-s + 8·23-s − 6·26-s + 2·29-s − 4·31-s + 5·32-s − 2·34-s + 2·37-s + 8·38-s − 6·41-s − 4·43-s + 8·46-s − 8·47-s + 6·52-s + 10·53-s + 2·58-s + 4·59-s + 2·61-s − 4·62-s + 7·64-s − 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.66·13-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 1.66·23-s − 1.17·26-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.328·37-s + 1.29·38-s − 0.937·41-s − 0.609·43-s + 1.17·46-s − 1.16·47-s + 0.832·52-s + 1.37·53-s + 0.262·58-s + 0.520·59-s + 0.256·61-s − 0.508·62-s + 7/8·64-s − 0.488·67-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 88.035088.0350
Root analytic conductor: 9.382709.38270
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 11025, ( :1/2), −1)(2,\ 11025,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad3 1 1
5 1 1
7 1 1
good2 1−T+pT2 1 - T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+18T+pT2 1 + 18 T + p 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

−16.72237721915346, −16.26165633230791, −15.35211768463677, −14.92358325157380, −14.60859600193107, −13.78590832522480, −13.47483419320935, −12.82028051186910, −12.26600392062564, −11.76157312455521, −11.20975416035185, −10.29224097768517, −9.587978706342278, −9.372712987930725, −8.564417779484274, −7.844750921391283, −7.048775273905974, −6.691265488473300, −5.476489495379934, −5.179223014180048, −4.701886696483178, −3.763628836400713, −3.070859664274092, −2.453991589086128, −1.127880531938779, 0, 1.127880531938779, 2.453991589086128, 3.070859664274092, 3.763628836400713, 4.701886696483178, 5.179223014180048, 5.476489495379934, 6.691265488473300, 7.048775273905974, 7.844750921391283, 8.564417779484274, 9.372712987930725, 9.587978706342278, 10.29224097768517, 11.20975416035185, 11.76157312455521, 12.26600392062564, 12.82028051186910, 13.47483419320935, 13.78590832522480, 14.60859600193107, 14.92358325157380, 15.35211768463677, 16.26165633230791, 16.72237721915346

Graph of the ZZ-function along the critical line