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Properties

Label 2-448-1.1-c3-0-11
Degree 22
Conductor 448448
Sign 11
Analytic cond. 26.432826.4328
Root an. cond. 5.141285.14128
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  − 2·3-s + 16·5-s + 7·7-s − 23·9-s + 24·11-s + 68·13-s − 32·15-s + 54·17-s − 46·19-s − 14·21-s − 176·23-s + 131·25-s + 100·27-s + 174·29-s + 116·31-s − 48·33-s + 112·35-s − 74·37-s − 136·39-s − 10·41-s − 480·43-s − 368·45-s + 572·47-s + 49·49-s − 108·51-s + 162·53-s + 384·55-s + ⋯
L(s)  = 1  − 0.384·3-s + 1.43·5-s + 0.377·7-s − 0.851·9-s + 0.657·11-s + 1.45·13-s − 0.550·15-s + 0.770·17-s − 0.555·19-s − 0.145·21-s − 1.59·23-s + 1.04·25-s + 0.712·27-s + 1.11·29-s + 0.672·31-s − 0.253·33-s + 0.540·35-s − 0.328·37-s − 0.558·39-s − 0.0380·41-s − 1.70·43-s − 1.21·45-s + 1.77·47-s + 1/7·49-s − 0.296·51-s + 0.419·53-s + 0.941·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 11
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 448, ( :3/2), 1)(2,\ 448,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.4041949522.404194952
L(12)L(\frac12) ≈\approx 2.4041949522.404194952
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
7 1−pT 1 - p T
good3 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
5 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
11 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
13 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
17 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
19 1+46T+p3T2 1 + 46 T + p^{3} T^{2}
23 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
29 1−6pT+p3T2 1 - 6 p T + p^{3} T^{2}
31 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
37 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
41 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
43 1+480T+p3T2 1 + 480 T + p^{3} T^{2}
47 1−572T+p3T2 1 - 572 T + p^{3} T^{2}
53 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
59 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
61 1−904T+p3T2 1 - 904 T + p^{3} T^{2}
67 1−660T+p3T2 1 - 660 T + p^{3} T^{2}
71 1+1024T+p3T2 1 + 1024 T + p^{3} T^{2}
73 1−770T+p3T2 1 - 770 T + p^{3} T^{2}
79 1−904T+p3T2 1 - 904 T + p^{3} T^{2}
83 1−682T+p3T2 1 - 682 T + p^{3} T^{2}
89 1+102T+p3T2 1 + 102 T + p^{3} T^{2}
97 1+218T+p3T2 1 + 218 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

−10.56480593280988231971019898375, −9.945878943104851608220139975777, −8.813979355129286989845266630100, −8.216718257510508179639488871985, −6.49918233833926407681722809876, −6.05012059571621196865816067982, −5.20110970311538089375129196469, −3.75046211898536870871056101514, −2.26641089135339627238958984135, −1.08599772617165914430570884843, 1.08599772617165914430570884843, 2.26641089135339627238958984135, 3.75046211898536870871056101514, 5.20110970311538089375129196469, 6.05012059571621196865816067982, 6.49918233833926407681722809876, 8.216718257510508179639488871985, 8.813979355129286989845266630100, 9.945878943104851608220139975777, 10.56480593280988231971019898375

Graph of the ZZ-function along the critical line