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Properties

Label 4-14e4-1.1-c3e2-0-0
Degree 44
Conductor 3841638416
Sign 11
Analytic cond. 133.734133.734
Root an. cond. 3.400643.40064
Motivic weight 33
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  − 4·3-s − 6·5-s + 27·9-s + 12·11-s − 164·13-s + 24·15-s + 30·17-s − 68·19-s − 216·23-s + 125·25-s − 260·27-s + 492·29-s + 112·31-s − 48·33-s − 110·37-s + 656·39-s − 492·41-s − 344·43-s − 162·45-s − 192·47-s − 120·51-s − 558·53-s − 72·55-s + 272·57-s − 540·59-s − 110·61-s + 984·65-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.536·5-s + 9-s + 0.328·11-s − 3.49·13-s + 0.413·15-s + 0.428·17-s − 0.821·19-s − 1.95·23-s + 25-s − 1.85·27-s + 3.15·29-s + 0.648·31-s − 0.253·33-s − 0.488·37-s + 2.69·39-s − 1.87·41-s − 1.21·43-s − 0.536·45-s − 0.595·47-s − 0.329·51-s − 1.44·53-s − 0.176·55-s + 0.632·57-s − 1.19·59-s − 0.230·61-s + 1.87·65-s + ⋯

Functional equation

Λ(s)=(38416s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(38416s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3841638416    =    24⋅742^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 133.734133.734
Root analytic conductor: 3.400643.40064
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 38416, ( :3/2,3/2), 1)(4,\ 38416,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.29780862480.2978086248
L(12)L(\frac12) ≈\approx 0.29780862480.2978086248
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
7 1 1
good3C22C_2^2 1+4T−11T2+4p3T3+p6T4 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
5C22C_2^2 1+6T−89T2+6p3T3+p6T4 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1−12T−1187T2−12p3T3+p6T4 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
13C2C_2 (1+82T+p3T2)2 ( 1 + 82 T + p^{3} T^{2} )^{2}
17C22C_2^2 1−30T−4013T2−30p3T3+p6T4 1 - 30 T - 4013 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+68T−2235T2+68p3T3+p6T4 1 + 68 T - 2235 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4}
23C22C_2^2 1+216T+34489T2+216p3T3+p6T4 1 + 216 T + 34489 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4}
29C2C_2 (1−246T+p3T2)2 ( 1 - 246 T + p^{3} T^{2} )^{2}
31C22C_2^2 1−112T−17247T2−112p3T3+p6T4 1 - 112 T - 17247 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4}
37C2C_2 (1−323T+p3T2)(1+433T+p3T2) ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} )
41C2C_2 (1+6pT+p3T2)2 ( 1 + 6 p T + p^{3} T^{2} )^{2}
43C2C_2 (1+4pT+p3T2)2 ( 1 + 4 p T + p^{3} T^{2} )^{2}
47C22C_2^2 1+192T−66959T2+192p3T3+p6T4 1 + 192 T - 66959 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4}
53C22C_2^2 1+558T+162487T2+558p3T3+p6T4 1 + 558 T + 162487 T^{2} + 558 p^{3} T^{3} + p^{6} T^{4}
59C22C_2^2 1+540T+86221T2+540p3T3+p6T4 1 + 540 T + 86221 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1+110T−214881T2+110p3T3+p6T4 1 + 110 T - 214881 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1+140T−281163T2+140p3T3+p6T4 1 + 140 T - 281163 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1+840T+p3T2)2 ( 1 + 840 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−550T−86517T2−550p3T3+p6T4 1 - 550 T - 86517 T^{2} - 550 p^{3} T^{3} + p^{6} T^{4}
79C22C_2^2 1−208T−449775T2−208p3T3+p6T4 1 - 208 T - 449775 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4}
83C2C_2 (1−516T+p3T2)2 ( 1 - 516 T + p^{3} T^{2} )^{2}
89C22C_2^2 1−1398T+1249435T2−1398p3T3+p6T4 1 - 1398 T + 1249435 T^{2} - 1398 p^{3} T^{3} + p^{6} T^{4}
97C2C_2 (1−1586T+p3T2)2 ( 1 - 1586 T + p^{3} T^{2} )^{2}
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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

−12.10990275456284331313979778812, −11.88053978802689795514574928802, −11.79188901751146939980168353871, −10.54989104233363636548041475150, −10.30662685768885459774046504083, −9.987021935702769805201489248938, −9.595615729319619324015713385917, −8.783763886731685448797685357591, −8.017871226687383828220941146861, −7.72392620067272347833698178259, −7.17032583369822417280083927410, −6.45917709767366171036123246480, −6.31931685194823523962653104055, −4.93372762172154909881859677125, −4.91866246350272275700469118060, −4.45366831255890056571636164624, −3.39910665228630714643905484122, −2.52773375524641331173771060948, −1.68793326661785399715810453616, −0.24248465276623332536090872727, 0.24248465276623332536090872727, 1.68793326661785399715810453616, 2.52773375524641331173771060948, 3.39910665228630714643905484122, 4.45366831255890056571636164624, 4.91866246350272275700469118060, 4.93372762172154909881859677125, 6.31931685194823523962653104055, 6.45917709767366171036123246480, 7.17032583369822417280083927410, 7.72392620067272347833698178259, 8.017871226687383828220941146861, 8.783763886731685448797685357591, 9.595615729319619324015713385917, 9.987021935702769805201489248938, 10.30662685768885459774046504083, 10.54989104233363636548041475150, 11.79188901751146939980168353871, 11.88053978802689795514574928802, 12.10990275456284331313979778812

Graph of the ZZ-function along the critical line