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Properties

Label 4-40e4-1.1-c1e2-0-16
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
Motivic weight 11
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  − 6·7-s + 2·9-s + 2·11-s + 4·13-s − 2·17-s − 6·19-s − 2·23-s + 14·29-s + 12·37-s + 8·43-s + 14·47-s + 18·49-s + 6·59-s − 2·61-s − 12·63-s + 8·67-s + 6·73-s − 12·77-s + 16·79-s − 5·81-s + 12·89-s − 24·91-s + 22·97-s + 4·99-s − 10·101-s − 10·103-s − 10·109-s + ⋯
L(s)  = 1  − 2.26·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 2.59·29-s + 1.97·37-s + 1.21·43-s + 2.04·47-s + 18/7·49-s + 0.781·59-s − 0.256·61-s − 1.51·63-s + 0.977·67-s + 0.702·73-s − 1.36·77-s + 1.80·79-s − 5/9·81-s + 1.27·89-s − 2.51·91-s + 2.23·97-s + 0.402·99-s − 0.995·101-s − 0.985·103-s − 0.957·109-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9455210211.945521021
L(12)L(\frac12) ≈\approx 1.9455210211.945521021
L(32)L(\frac{3}{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
5 1 1
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1−4T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} )
31C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
59C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−10T+pT2)(1+12T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−162T2+p2T4 1 - 162 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C22C_2^2 1−22T+242T2−22pT3+p2T4 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} 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

−9.559984904623230768120056510743, −9.202777442018719474560797104627, −8.882847590088192414502997957351, −8.626973638120059890081391121437, −7.86882957691107954305422945914, −7.79877819103494708854545042448, −6.81502325209710488933902103630, −6.80769583172628534747188827900, −6.29016806193117293938364217018, −6.26327312788276781106289806141, −5.78312796995093100510417287078, −5.04329672371956769339323219259, −4.35000627073342753358568227050, −4.05001816239171832702076295634, −3.82840733122396495336882871672, −3.16192918225020170956696752608, −2.53961247208360371430219396653, −2.31408596923977728867184232745, −1.10895467926714307885772631133, −0.64075104863877162412199913489, 0.64075104863877162412199913489, 1.10895467926714307885772631133, 2.31408596923977728867184232745, 2.53961247208360371430219396653, 3.16192918225020170956696752608, 3.82840733122396495336882871672, 4.05001816239171832702076295634, 4.35000627073342753358568227050, 5.04329672371956769339323219259, 5.78312796995093100510417287078, 6.26327312788276781106289806141, 6.29016806193117293938364217018, 6.80769583172628534747188827900, 6.81502325209710488933902103630, 7.79877819103494708854545042448, 7.86882957691107954305422945914, 8.626973638120059890081391121437, 8.882847590088192414502997957351, 9.202777442018719474560797104627, 9.559984904623230768120056510743

Graph of the ZZ-function along the critical line