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Properties

Label 4-416e2-1.1-c2e2-0-0
Degree 44
Conductor 173056173056
Sign 11
Analytic cond. 128.486128.486
Root an. cond. 3.366773.36677
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s − 5·7-s + 9·9-s − 21·11-s − 26·13-s − 4·15-s + 25·17-s − 21·19-s − 5·21-s − 15·23-s − 38·25-s + 26·27-s − 39·29-s − 20·31-s − 21·33-s + 20·35-s − 61·37-s − 26·39-s + 3·41-s + 41·43-s − 36·45-s − 164·47-s + 49·49-s + 25·51-s + 84·55-s − 21·57-s + ⋯
L(s)  = 1  + 1/3·3-s − 4/5·5-s − 5/7·7-s + 9-s − 1.90·11-s − 2·13-s − 0.266·15-s + 1.47·17-s − 1.10·19-s − 0.238·21-s − 0.652·23-s − 1.51·25-s + 0.962·27-s − 1.34·29-s − 0.645·31-s − 0.636·33-s + 4/7·35-s − 1.64·37-s − 2/3·39-s + 3/41·41-s + 0.953·43-s − 4/5·45-s − 3.48·47-s + 49-s + 0.490·51-s + 1.52·55-s − 0.368·57-s + ⋯

Functional equation

Λ(s)=(173056s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(173056s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 173056173056    =    210⋅1322^{10} \cdot 13^{2}
Sign: 11
Analytic conductor: 128.486128.486
Root analytic conductor: 3.366773.36677
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 173056, ( :1,1), 1)(4,\ 173056,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.031513177090.03151317709
L(12)L(\frac12) ≈\approx 0.031513177090.03151317709
L(2)L(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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1−T−8T2−p2T3+p4T4 1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4}
5C2C_2 (1+2T+p2T2)2 ( 1 + 2 T + p^{2} T^{2} )^{2}
7C22C_2^2 1+5T−24T2+5p2T3+p4T4 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4}
11C22C_2^2 1+21T+268T2+21p2T3+p4T4 1 + 21 T + 268 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4}
17C22C_2^2 1−25T+336T2−25p2T3+p4T4 1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4}
19C22C_2^2 1+21T+508T2+21p2T3+p4T4 1 + 21 T + 508 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4}
23C22C_2^2 1+15T+604T2+15p2T3+p4T4 1 + 15 T + 604 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4}
29C22C_2^2 1+39T+1348T2+39p2T3+p4T4 1 + 39 T + 1348 T^{2} + 39 p^{2} T^{3} + p^{4} T^{4}
31C2C_2 (1+10T+p2T2)2 ( 1 + 10 T + p^{2} T^{2} )^{2}
37C22C_2^2 1+61T+2352T2+61p2T3+p4T4 1 + 61 T + 2352 T^{2} + 61 p^{2} T^{3} + p^{4} T^{4}
41C22C_2^2 1−3T+1684T2−3p2T3+p4T4 1 - 3 T + 1684 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4}
43C22C_2^2 1−41T−168T2−41p2T3+p4T4 1 - 41 T - 168 T^{2} - 41 p^{2} T^{3} + p^{4} T^{4}
47C2C_2 (1+82T+p2T2)2 ( 1 + 82 T + p^{2} T^{2} )^{2}
53C22C_2^2 1−3890T2+p4T4 1 - 3890 T^{2} + p^{4} T^{4}
59C22C_2^2 1+93T+6364T2+93p2T3+p4T4 1 + 93 T + 6364 T^{2} + 93 p^{2} T^{3} + p^{4} T^{4}
61C22C_2^2 1−105T+7396T2−105p2T3+p4T4 1 - 105 T + 7396 T^{2} - 105 p^{2} T^{3} + p^{4} T^{4}
67C22C_2^2 1+21T+4636T2+21p2T3+p4T4 1 + 21 T + 4636 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4}
71C22C_2^2 1+29T−4200T2+29p2T3+p4T4 1 + 29 T - 4200 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4}
73C22C_2^2 1−8930T2+p4T4 1 - 8930 T^{2} + p^{4} T^{4}
79C22C_2^2 1+3070T2+p4T4 1 + 3070 T^{2} + p^{4} T^{4}
83C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
89C22C_2^2 1+69T+9508T2+69p2T3+p4T4 1 + 69 T + 9508 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4}
97C22C_2^2 1−3T+9412T2−3p2T3+p4T4 1 - 3 T + 9412 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.29040295736550283285667293205, −10.44135609810443481103088926613, −10.33343532601651005766173931593, −9.885735228627100200078535242408, −9.640426674169075849771594702961, −9.039102680267200962145239292550, −8.183338022836763910977438536758, −7.906812469792846892192862485741, −7.68297991626566231873232377160, −7.08598502259984240722640107972, −6.85992885243061578066788485321, −5.68731308123808046635422054628, −5.64998744928129489452084170215, −4.73252169514818720679078227558, −4.48109302048125822098895603557, −3.49201665856261041914624576306, −3.32560028170798888975323935755, −2.32577728057659925821661903861, −1.88035305647597642884520458780, −0.06938897454619178712173752298, 0.06938897454619178712173752298, 1.88035305647597642884520458780, 2.32577728057659925821661903861, 3.32560028170798888975323935755, 3.49201665856261041914624576306, 4.48109302048125822098895603557, 4.73252169514818720679078227558, 5.64998744928129489452084170215, 5.68731308123808046635422054628, 6.85992885243061578066788485321, 7.08598502259984240722640107972, 7.68297991626566231873232377160, 7.906812469792846892192862485741, 8.183338022836763910977438536758, 9.039102680267200962145239292550, 9.640426674169075849771594702961, 9.885735228627100200078535242408, 10.33343532601651005766173931593, 10.44135609810443481103088926613, 11.29040295736550283285667293205

Graph of the ZZ-function along the critical line