↑

Properties

Label 2-105e2-1.1-c1-0-6
Degree 22
Conductor 1102511025
Sign 11
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 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s + 2·13-s − 16-s − 2·17-s − 6·19-s − 3·23-s + 2·26-s − 7·29-s − 2·31-s + 5·32-s − 2·34-s + 8·37-s − 6·38-s + 5·41-s − 7·43-s − 3·46-s − 2·52-s + 6·53-s − 7·58-s + 10·59-s − 7·61-s − 2·62-s + 7·64-s + 5·67-s + 2·68-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.554·13-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.625·23-s + 0.392·26-s − 1.29·29-s − 0.359·31-s + 0.883·32-s − 0.342·34-s + 1.31·37-s − 0.973·38-s + 0.780·41-s − 1.06·43-s − 0.442·46-s − 0.277·52-s + 0.824·53-s − 0.919·58-s + 1.30·59-s − 0.896·61-s − 0.254·62-s + 7/8·64-s + 0.610·67-s + 0.242·68-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: 11
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: 00
Selberg data: (2, 11025, ( :1/2), 1)(2,\ 11025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6280208101.628020810
L(12)L(\frac12) ≈\approx 1.6280208101.628020810
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 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−11T+pT2 1 - 11 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−16.61926358749564, −15.74483563433124, −15.20945036879121, −14.69081782046047, −14.30815422810820, −13.46199092239385, −13.11040563926872, −12.76398231623780, −11.95432036059858, −11.40127309615421, −10.78649577153838, −10.08838712665351, −9.369696018187384, −8.860456871585890, −8.285201277706972, −7.608009193149179, −6.646885593873036, −6.119125883668494, −5.551567360977953, −4.759136833407733, −4.062283761536627, −3.698303783140454, −2.660833641933034, −1.864649498893109, −0.5168829589861634, 0.5168829589861634, 1.864649498893109, 2.660833641933034, 3.698303783140454, 4.062283761536627, 4.759136833407733, 5.551567360977953, 6.119125883668494, 6.646885593873036, 7.608009193149179, 8.285201277706972, 8.860456871585890, 9.369696018187384, 10.08838712665351, 10.78649577153838, 11.40127309615421, 11.95432036059858, 12.76398231623780, 13.11040563926872, 13.46199092239385, 14.30815422810820, 14.69081782046047, 15.20945036879121, 15.74483563433124, 16.61926358749564

Graph of the ZZ-function along the critical line