↑

Properties

Label 2-704-11.10-c0-0-0
Degree 22
Conductor 704704
Sign 11
Analytic cond. 0.3513410.351341
Root an. cond. 0.5927400.592740
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 11-s − 15-s + 23-s + 27-s + 31-s − 33-s + 37-s − 2·47-s + 49-s − 2·53-s + 55-s − 59-s − 67-s − 69-s + 71-s − 81-s − 89-s − 93-s − 97-s − 2·103-s − 111-s − 113-s + 115-s + ⋯
L(s)  = 1  − 3-s + 5-s + 11-s − 15-s + 23-s + 27-s + 31-s − 33-s + 37-s − 2·47-s + 49-s − 2·53-s + 55-s − 59-s − 67-s − 69-s + 71-s − 81-s − 89-s − 93-s − 97-s − 2·103-s − 111-s − 113-s + 115-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: 11
Analytic conductor: 0.3513410.351341
Root analytic conductor: 0.5927400.592740
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ704(65,⋅)\chi_{704} (65, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 704, ( :0), 1)(2,\ 704,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83421093240.8342109324
L(12)L(\frac12) ≈\approx 0.83421093240.8342109324
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 1
11 1−T 1 - T
good3 1+T+T2 1 + T + T^{2}
5 1−T+T2 1 - T + T^{2}
7 (1−T)(1+T) ( 1 - T )( 1 + T )
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 1−T+T2 1 - T + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1−T+T2 1 - T + T^{2}
37 1−T+T2 1 - T + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1+T)2 ( 1 + T )^{2}
53 (1+T)2 ( 1 + T )^{2}
59 1+T+T2 1 + T + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 1+T+T2 1 + T + T^{2}
71 1−T+T2 1 - T + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 1+T+T2 1 + T + T^{2}
97 1+T+T2 1 + T + 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

−10.76986502874820360446020620007, −9.775517195333216833439651901402, −9.189706642100871132128751495689, −8.087796887651203646924886576649, −6.71158914396420532070739687727, −6.24407165765924451166630290208, −5.38319360534408517236765833403, −4.46317942736026664517383129516, −2.93434243358027300820327152618, −1.39587762472223235785737155063, 1.39587762472223235785737155063, 2.93434243358027300820327152618, 4.46317942736026664517383129516, 5.38319360534408517236765833403, 6.24407165765924451166630290208, 6.71158914396420532070739687727, 8.087796887651203646924886576649, 9.189706642100871132128751495689, 9.775517195333216833439651901402, 10.76986502874820360446020620007

Graph of the ZZ-function along the critical line