↑

Properties

Label 2-8400-1.1-c1-0-6
Degree 22
Conductor 84008400
Sign 11
Analytic cond. 67.074367.0743
Root an. cond. 8.189898.18989
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  − 3-s + 7-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 2·19-s − 21-s − 6·23-s − 27-s + 6·29-s − 6·31-s + 4·33-s + 4·37-s + 2·39-s − 4·43-s + 4·47-s + 49-s + 2·51-s + 2·53-s − 2·57-s − 4·59-s − 2·61-s + 63-s − 12·67-s + 6·69-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.696·33-s + 0.657·37-s + 0.320·39-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.264·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.722·69-s + 0.949·71-s + ⋯

Functional equation

Λ(s)=(8400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 84008400    =    24⋅3⋅52⋅72^{4} \cdot 3 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 67.074367.0743
Root analytic conductor: 8.189898.18989
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8400, ( :1/2), 1)(2,\ 8400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0508986571.050898657
L(12)L(\frac12) ≈\approx 1.0508986571.050898657
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+T 1 + T
5 1 1
7 1−T 1 - T
good11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−2T+pT2 1 - 2 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+6T+pT2 1 + 6 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−16T+pT2 1 - 16 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−7.56279277731945914272912400131, −7.35048406859826782523602071262, −6.23306806472184064493666247970, −5.78489214933156331783686829675, −4.89697965516847316969092004032, −4.57114548767155620108541318156, −3.50101908861801476802011770116, −2.55181171513341671521163672538, −1.79768067173493383952295287520, −0.50273118136580517687805037565, 0.50273118136580517687805037565, 1.79768067173493383952295287520, 2.55181171513341671521163672538, 3.50101908861801476802011770116, 4.57114548767155620108541318156, 4.89697965516847316969092004032, 5.78489214933156331783686829675, 6.23306806472184064493666247970, 7.35048406859826782523602071262, 7.56279277731945914272912400131

Graph of the ZZ-function along the critical line