↑

Properties

Label 2-9702-1.1-c1-0-130
Degree 22
Conductor 97029702
Sign −1-1
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2.82·5-s + 8-s − 2.82·10-s − 11-s + 4.24·13-s + 16-s + 2.82·17-s + 1.41·19-s − 2.82·20-s − 22-s − 6·23-s + 3.00·25-s + 4.24·26-s − 8·29-s − 1.41·31-s + 32-s + 2.82·34-s − 6·37-s + 1.41·38-s − 2.82·40-s − 8.48·41-s + 10·43-s − 44-s − 6·46-s + 7.07·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.26·5-s + 0.353·8-s − 0.894·10-s − 0.301·11-s + 1.17·13-s + 0.250·16-s + 0.685·17-s + 0.324·19-s − 0.632·20-s − 0.213·22-s − 1.25·23-s + 0.600·25-s + 0.832·26-s − 1.48·29-s − 0.254·31-s + 0.176·32-s + 0.485·34-s − 0.986·37-s + 0.229·38-s − 0.447·40-s − 1.32·41-s + 1.52·43-s − 0.150·44-s − 0.884·46-s + 1.03·47-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: −1-1
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9702, ( :1/2), −1)(2,\ 9702,\ (\ :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)
bad2 1−T 1 - T
3 1 1
7 1 1
11 1+T 1 + T
good5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
13 1−4.24T+13T2 1 - 4.24T + 13T^{2}
17 1−2.82T+17T2 1 - 2.82T + 17T^{2}
19 1−1.41T+19T2 1 - 1.41T + 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1+1.41T+31T2 1 + 1.41T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+8.48T+41T2 1 + 8.48T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1−7.07T+47T2 1 - 7.07T + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−14.1T+59T2 1 - 14.1T + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−8.48T+73T2 1 - 8.48T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−7.07T+83T2 1 - 7.07T + 83T^{2}
89 1+18.3T+89T2 1 + 18.3T + 89T^{2}
97 1+1.41T+97T2 1 + 1.41T + 97T^{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

−7.41778712323278118655027061120, −6.66121329136082558814821834478, −5.78311394837978738967266114018, −5.37713483173816357730568533469, −4.36089896266003207325585601039, −3.68671553357116794443664839923, −3.48578985938953674834555666432, −2.30936118986711341920204592261, −1.27868173372719731314821292033, 0, 1.27868173372719731314821292033, 2.30936118986711341920204592261, 3.48578985938953674834555666432, 3.68671553357116794443664839923, 4.36089896266003207325585601039, 5.37713483173816357730568533469, 5.78311394837978738967266114018, 6.66121329136082558814821834478, 7.41778712323278118655027061120

Graph of the ZZ-function along the critical line