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Properties

Label 2-336-1.1-c3-0-0
Degree 22
Conductor 336336
Sign 11
Analytic cond. 19.824619.8246
Root an. cond. 4.452484.45248
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 16·5-s − 7·7-s + 9·9-s + 18·11-s − 54·13-s + 48·15-s − 128·17-s − 52·19-s + 21·21-s + 202·23-s + 131·25-s − 27·27-s + 302·29-s + 200·31-s − 54·33-s + 112·35-s − 150·37-s + 162·39-s + 172·41-s − 164·43-s − 144·45-s + 460·47-s + 49·49-s + 384·51-s − 190·53-s − 288·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.43·5-s − 0.377·7-s + 1/3·9-s + 0.493·11-s − 1.15·13-s + 0.826·15-s − 1.82·17-s − 0.627·19-s + 0.218·21-s + 1.83·23-s + 1.04·25-s − 0.192·27-s + 1.93·29-s + 1.15·31-s − 0.284·33-s + 0.540·35-s − 0.666·37-s + 0.665·39-s + 0.655·41-s − 0.581·43-s − 0.477·45-s + 1.42·47-s + 1/7·49-s + 1.05·51-s − 0.492·53-s − 0.706·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 19.824619.8246
Root analytic conductor: 4.452484.45248
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 336, ( :3/2), 1)(2,\ 336,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.74098061010.7409806101
L(12)L(\frac12) ≈\approx 0.74098061010.7409806101
L(52)L(\frac{5}{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+pT 1 + p T
7 1+pT 1 + p T
good5 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
13 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
17 1+128T+p3T2 1 + 128 T + p^{3} T^{2}
19 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
23 1−202T+p3T2 1 - 202 T + p^{3} T^{2}
29 1−302T+p3T2 1 - 302 T + p^{3} T^{2}
31 1−200T+p3T2 1 - 200 T + p^{3} T^{2}
37 1+150T+p3T2 1 + 150 T + p^{3} T^{2}
41 1−172T+p3T2 1 - 172 T + p^{3} T^{2}
43 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
47 1−460T+p3T2 1 - 460 T + p^{3} T^{2}
53 1+190T+p3T2 1 + 190 T + p^{3} T^{2}
59 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
61 1−622T+p3T2 1 - 622 T + p^{3} T^{2}
67 1+744T+p3T2 1 + 744 T + p^{3} T^{2}
71 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
73 1−742T+p3T2 1 - 742 T + p^{3} T^{2}
79 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
83 1−228T+p3T2 1 - 228 T + p^{3} T^{2}
89 1+116T+p3T2 1 + 116 T + p^{3} T^{2}
97 1+554T+p3T2 1 + 554 T + p^{3} T^{2}
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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

−11.20876583832361548647321631046, −10.45478755097439234300758133236, −9.173161647440508266531533252313, −8.303044130929428967487671027956, −7.05245144103255058414809493545, −6.59900792753048558806810860192, −4.84938853627883719688575040971, −4.22339317529329783087355723542, −2.74849238853855545214653975714, −0.58171754790662278082856459855, 0.58171754790662278082856459855, 2.74849238853855545214653975714, 4.22339317529329783087355723542, 4.84938853627883719688575040971, 6.59900792753048558806810860192, 7.05245144103255058414809493545, 8.303044130929428967487671027956, 9.173161647440508266531533252313, 10.45478755097439234300758133236, 11.20876583832361548647321631046

Graph of the ZZ-function along the critical line