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Properties

Label 4-2736e2-1.1-c1e2-0-7
Degree 44
Conductor 74856967485696
Sign 11
Analytic cond. 477.294477.294
Root an. cond. 4.674084.67408
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  − 4·7-s + 2·19-s + 8·25-s + 20·29-s − 20·41-s − 24·43-s − 2·49-s − 20·53-s − 24·59-s + 16·61-s − 16·71-s + 12·73-s − 12·89-s − 16·107-s − 12·113-s + 4·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.458·19-s + 8/5·25-s + 3.71·29-s − 3.12·41-s − 3.65·43-s − 2/7·49-s − 2.74·53-s − 3.12·59-s + 2.04·61-s − 1.89·71-s + 1.40·73-s − 1.27·89-s − 1.54·107-s − 1.12·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.38·169-s + ⋯

Functional equation

Λ(s)=(7485696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7485696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 74856967485696    =    28⋅34⋅1922^{8} \cdot 3^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 477.294477.294
Root analytic conductor: 4.674084.67408
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 7485696, ( :1/2,1/2), 1)(4,\ 7485696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.70141182200.7014118220
L(12)L(\frac12) ≈\approx 0.70141182200.7014118220
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
3 1 1
19C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good5C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
13C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
17C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
23C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
31C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
37C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
47C22C_2^2 1−92T2+p2T4 1 - 92 T^{2} + p^{2} T^{4}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C22C_2^2 1+66T2+p2T4 1 + 66 T^{2} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
83C22C_2^2 1−148T2+p2T4 1 - 148 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−122T2+p2T4 1 - 122 T^{2} + 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

−9.347453729947505209716856214190, −8.414066668104710553780849846131, −8.388607560446760135603325460951, −8.097423073832669930039587754061, −7.47295660411565597160686734721, −6.71360108475616984799355836452, −6.68879116065859589580559759584, −6.51021849318124771537312908184, −6.32466854167710440351249863080, −5.36623782528449259751731534900, −5.11515380299465485510014742758, −4.65057334902231568529829007215, −4.55508378514584626340535146117, −3.44589469951260031632651024228, −3.35727701118365112113449531692, −2.97076155210468261316761445450, −2.71705027548671942655146982522, −1.53610189667986003881240686396, −1.41449687756811312408534331126, −0.27512453030734254081306282151, 0.27512453030734254081306282151, 1.41449687756811312408534331126, 1.53610189667986003881240686396, 2.71705027548671942655146982522, 2.97076155210468261316761445450, 3.35727701118365112113449531692, 3.44589469951260031632651024228, 4.55508378514584626340535146117, 4.65057334902231568529829007215, 5.11515380299465485510014742758, 5.36623782528449259751731534900, 6.32466854167710440351249863080, 6.51021849318124771537312908184, 6.68879116065859589580559759584, 6.71360108475616984799355836452, 7.47295660411565597160686734721, 8.097423073832669930039587754061, 8.388607560446760135603325460951, 8.414066668104710553780849846131, 9.347453729947505209716856214190

Graph of the ZZ-function along the critical line