↑

Properties

Label 2-29988-1.1-c1-0-17
Degree 22
Conductor 2998829988
Sign 11
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
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  + 5·13-s + 17-s − 19-s + 6·23-s − 5·25-s + 6·29-s + 5·31-s − 7·37-s + 6·41-s − 43-s + 6·47-s − 6·53-s − 10·61-s + 5·67-s + 6·71-s − 73-s − 79-s + 6·83-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.38·13-s + 0.242·17-s − 0.229·19-s + 1.25·23-s − 25-s + 1.11·29-s + 0.898·31-s − 1.15·37-s + 0.937·41-s − 0.152·43-s + 0.875·47-s − 0.824·53-s − 1.28·61-s + 0.610·67-s + 0.712·71-s − 0.117·73-s − 0.112·79-s + 0.658·83-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 29988, ( :1/2), 1)(2,\ 29988,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7147842632.714784263
L(12)L(\frac12) ≈\approx 2.7147842632.714784263
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 1
3 1 1
7 1 1
17 1−T 1 - T
good5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+T+pT2 1 + T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+pT2 1 + 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

−15.33236615400928, −14.54643503542151, −13.93072979824529, −13.68758089424254, −13.04908860126643, −12.47615809826695, −11.98722134524599, −11.29042908200112, −10.93251389994541, −10.31251666988736, −9.830441674730393, −8.912816453334113, −8.812186304103534, −8.002536999748912, −7.543627025770976, −6.714773422770044, −6.272823633245713, −5.723819063910142, −4.950067485731543, −4.382196567105888, −3.603526083159051, −3.110623659628546, −2.252286883803925, −1.370665697836646, −0.6857396848635597, 0.6857396848635597, 1.370665697836646, 2.252286883803925, 3.110623659628546, 3.603526083159051, 4.382196567105888, 4.950067485731543, 5.723819063910142, 6.272823633245713, 6.714773422770044, 7.543627025770976, 8.002536999748912, 8.812186304103534, 8.912816453334113, 9.830441674730393, 10.31251666988736, 10.93251389994541, 11.29042908200112, 11.98722134524599, 12.47615809826695, 13.04908860126643, 13.68758089424254, 13.93072979824529, 14.54643503542151, 15.33236615400928

Graph of the ZZ-function along the critical line