↑

Properties

Label 2-56e2-1.1-c1-0-4
Degree 22
Conductor 31363136
Sign 11
Analytic cond. 25.041025.0410
Root an. cond. 5.004105.00410
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 4·13-s − 6·17-s + 2·19-s − 5·25-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + 12·51-s − 6·53-s − 4·57-s − 6·59-s + 8·61-s + 4·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s − 6·83-s − 12·87-s + 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1.68·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s + 1.02·61-s + 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s − 0.658·83-s − 1.28·87-s + 0.635·89-s + ⋯

Functional equation

Λ(s)=(3136s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3136s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31363136    =    26⋅722^{6} \cdot 7^{2}
Sign: 11
Analytic conductor: 25.041025.0410
Root analytic conductor: 5.004105.00410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3136, ( :1/2), 1)(2,\ 3136,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.70850486970.7085048697
L(12)L(\frac12) ≈\approx 0.70850486970.7085048697
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
7 1 1
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−8.678731346329368257431074943309, −7.895164678888831912291936607752, −6.90531576344093792893651903833, −6.50364338668347614528670710226, −5.58920639964454334454555282661, −4.91171613213171916096600238802, −4.29390759056738033972615396659, −2.99977174751051762376143721143, −1.99280794987984862469967531446, −0.51891811197874657341665639572, 0.51891811197874657341665639572, 1.99280794987984862469967531446, 2.99977174751051762376143721143, 4.29390759056738033972615396659, 4.91171613213171916096600238802, 5.58920639964454334454555282661, 6.50364338668347614528670710226, 6.90531576344093792893651903833, 7.895164678888831912291936607752, 8.678731346329368257431074943309

Graph of the ZZ-function along the critical line