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Properties

Label 2-5408-1.1-c1-0-80
Degree 22
Conductor 54085408
Sign −1-1
Analytic cond. 43.183043.1830
Root an. cond. 6.571386.57138
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 3·9-s + 2·17-s + 11·25-s + 10·29-s + 12·37-s − 8·41-s + 12·45-s − 7·49-s + 14·53-s − 10·61-s − 16·73-s + 9·81-s − 8·85-s − 16·89-s − 8·97-s + 2·101-s − 20·109-s + 14·113-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 0.485·17-s + 11/5·25-s + 1.85·29-s + 1.97·37-s − 1.24·41-s + 1.78·45-s − 49-s + 1.92·53-s − 1.28·61-s − 1.87·73-s + 81-s − 0.867·85-s − 1.69·89-s − 0.812·97-s + 0.199·101-s − 1.91·109-s + 1.31·113-s + ⋯

Functional equation

Λ(s)=(5408s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5408s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54085408    =    25⋅1322^{5} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 43.183043.1830
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5408, ( :1/2), −1)(2,\ 5408,\ (\ :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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
13 1 1
good3 1+pT2 1 + p T^{2} 1.3.a
5 1+4T+pT2 1 + 4 T + p T^{2} 1.5.e
7 1+pT2 1 + p T^{2} 1.7.a
11 1+pT2 1 + p T^{2} 1.11.a
17 1−2T+pT2 1 - 2 T + p T^{2} 1.17.ac
19 1+pT2 1 + p T^{2} 1.19.a
23 1+pT2 1 + p T^{2} 1.23.a
29 1−10T+pT2 1 - 10 T + p T^{2} 1.29.ak
31 1+pT2 1 + p T^{2} 1.31.a
37 1−12T+pT2 1 - 12 T + p T^{2} 1.37.am
41 1+8T+pT2 1 + 8 T + p T^{2} 1.41.i
43 1+pT2 1 + p T^{2} 1.43.a
47 1+pT2 1 + p T^{2} 1.47.a
53 1−14T+pT2 1 - 14 T + p T^{2} 1.53.ao
59 1+pT2 1 + p T^{2} 1.59.a
61 1+10T+pT2 1 + 10 T + p T^{2} 1.61.k
67 1+pT2 1 + p T^{2} 1.67.a
71 1+pT2 1 + p T^{2} 1.71.a
73 1+16T+pT2 1 + 16 T + p T^{2} 1.73.q
79 1+pT2 1 + p T^{2} 1.79.a
83 1+pT2 1 + p T^{2} 1.83.a
89 1+16T+pT2 1 + 16 T + p T^{2} 1.89.q
97 1+8T+pT2 1 + 8 T + p T^{2} 1.97.i
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   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

−7.988612977350406093974206442013, −7.21424279733010872572454801516, −6.49364081247895273094607913476, −5.62302150895633800615973468947, −4.70807460073308666993587405749, −4.14113725726325628790675953937, −3.21139015568415693943466713575, −2.72602212485500455177659066469, −1.04864477909657256988988373490, 0, 1.04864477909657256988988373490, 2.72602212485500455177659066469, 3.21139015568415693943466713575, 4.14113725726325628790675953937, 4.70807460073308666993587405749, 5.62302150895633800615973468947, 6.49364081247895273094607913476, 7.21424279733010872572454801516, 7.988612977350406093974206442013

Graph of the ZZ-function along the critical line