↑

Properties

Label 2-9702-1.1-c1-0-103
Degree 22
Conductor 97029702
Sign 11
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
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 + 2·5-s + 8-s + 2·10-s + 11-s + 4·13-s + 16-s + 2·17-s + 6·19-s + 2·20-s + 22-s + 2·23-s − 25-s + 4·26-s + 2·29-s + 8·31-s + 32-s + 2·34-s − 2·37-s + 6·38-s + 2·40-s − 2·41-s − 2·43-s + 44-s + 2·46-s + 2·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 1/4·16-s + 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.213·22-s + 0.417·23-s − 1/5·25-s + 0.784·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.973·38-s + 0.316·40-s − 0.312·41-s − 0.304·43-s + 0.150·44-s + 0.294·46-s + 0.291·47-s + ⋯

Functional equation

Λ(s)=(9702s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9702s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97029702    =    2⋅32⋅72⋅112 \cdot 3^{2} \cdot 7^{2} \cdot 11
Sign: 11
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9702, ( :1/2), 1)(2,\ 9702,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.0620762185.062076218
L(12)L(\frac12) ≈\approx 5.0620762185.062076218
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)
bad2 1−T 1 - T
3 1 1
7 1 1
11 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
13 1−4T+pT2 1 - 4 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 1−2T+pT2 1 - 2 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−7.55681813847005856187427135991, −6.79923493510247629545251393096, −6.15158471182841855400365242253, −5.69276000012681049628626494586, −5.00735722361764893107385149224, −4.23232022692161449850766373388, −3.33365417041840785728112437626, −2.83066461663364441278499665394, −1.69008397864184290335637215528, −1.06738870446855368402014094401, 1.06738870446855368402014094401, 1.69008397864184290335637215528, 2.83066461663364441278499665394, 3.33365417041840785728112437626, 4.23232022692161449850766373388, 5.00735722361764893107385149224, 5.69276000012681049628626494586, 6.15158471182841855400365242253, 6.79923493510247629545251393096, 7.55681813847005856187427135991

Graph of the ZZ-function along the critical line