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Properties

Label 2-336-1.1-c3-0-9
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 + 18·5-s − 7·7-s + 9·9-s + 72·11-s − 34·13-s + 54·15-s + 6·17-s − 92·19-s − 21·21-s + 180·23-s + 199·25-s + 27·27-s − 114·29-s − 56·31-s + 216·33-s − 126·35-s − 34·37-s − 102·39-s + 6·41-s − 164·43-s + 162·45-s − 168·47-s + 49·49-s + 18·51-s + 654·53-s + 1.29e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.60·5-s − 0.377·7-s + 1/3·9-s + 1.97·11-s − 0.725·13-s + 0.929·15-s + 0.0856·17-s − 1.11·19-s − 0.218·21-s + 1.63·23-s + 1.59·25-s + 0.192·27-s − 0.729·29-s − 0.324·31-s + 1.13·33-s − 0.608·35-s − 0.151·37-s − 0.418·39-s + 0.0228·41-s − 0.581·43-s + 0.536·45-s − 0.521·47-s + 1/7·49-s + 0.0494·51-s + 1.69·53-s + 3.17·55-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 3.1676907383.167690738
L(12)L(\frac12) ≈\approx 3.1676907383.167690738
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 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
11 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
19 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
23 1−180T+p3T2 1 - 180 T + p^{3} T^{2}
29 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
31 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
37 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
41 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
43 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
47 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
53 1−654T+p3T2 1 - 654 T + p^{3} T^{2}
59 1−492T+p3T2 1 - 492 T + p^{3} T^{2}
61 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
67 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
71 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
73 1−1010T+p3T2 1 - 1010 T + p^{3} T^{2}
79 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
83 1+228T+p3T2 1 + 228 T + p^{3} T^{2}
89 1−390T+p3T2 1 - 390 T + p^{3} T^{2}
97 1+70T+p3T2 1 + 70 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

−10.97063371379144635756312674025, −9.887411038356644898325913128793, −9.309185910214281596244060836286, −8.704711987937490937472325150768, −6.98841003189115270424790114680, −6.44031836747561565089771180367, −5.23966758715370911869091302667, −3.85577483848919244336909269014, −2.46611696453234412923932194191, −1.37817848240816738073129413396, 1.37817848240816738073129413396, 2.46611696453234412923932194191, 3.85577483848919244336909269014, 5.23966758715370911869091302667, 6.44031836747561565089771180367, 6.98841003189115270424790114680, 8.704711987937490937472325150768, 9.309185910214281596244060836286, 9.887411038356644898325913128793, 10.97063371379144635756312674025

Graph of the ZZ-function along the critical line