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Properties

Label 4-1134e2-1.1-c1e2-0-14
Degree 44
Conductor 12859561285956
Sign 11
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
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  − 2·2-s + 3·4-s + 5-s − 5·7-s − 4·8-s − 2·10-s + 5·11-s + 10·14-s + 5·16-s − 4·17-s − 8·19-s + 3·20-s − 10·22-s − 4·23-s + 5·25-s − 15·28-s − 5·29-s + 6·31-s − 6·32-s + 8·34-s − 5·35-s + 4·37-s + 16·38-s − 4·40-s − 2·43-s + 15·44-s + 8·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.447·5-s − 1.88·7-s − 1.41·8-s − 0.632·10-s + 1.50·11-s + 2.67·14-s + 5/4·16-s − 0.970·17-s − 1.83·19-s + 0.670·20-s − 2.13·22-s − 0.834·23-s + 25-s − 2.83·28-s − 0.928·29-s + 1.07·31-s − 1.06·32-s + 1.37·34-s − 0.845·35-s + 0.657·37-s + 2.59·38-s − 0.632·40-s − 0.304·43-s + 2.26·44-s + 1.17·46-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: 11
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1285956, ( :1/2,1/2), 1)(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54369234440.5436923444
L(12)L(\frac12) ≈\approx 0.54369234440.5436923444
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3 1 1
7C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good5C22C_2^2 1−T−4T2−pT3+p2T4 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−5T+14T2−5pT3+p2T4 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
17C22C_2^2 1+4T−T2+4pT3+p2T4 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4}
19C2C_2 (1+T+pT2)(1+7T+pT2) ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} )
23C22C_2^2 1+4T−7T2+4pT3+p2T4 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+5T−4T2+5pT3+p2T4 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4}
31C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
37C22C_2^2 1−4T−21T2−4pT3+p2T4 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C22C_2^2 1+2T−39T2+2pT3+p2T4 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4}
47C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
53C22C_2^2 1+9T+28T2+9pT3+p2T4 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4}
59C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
61C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
73C2C_2 (1−7T+pT2)(1+17T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
83C22C_2^2 1+7T−34T2+7pT3+p2T4 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+7T−48T2+7pT3+p2T4 1 + 7 T - 48 T^{2} + 7 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

−9.840462188827719874446867679209, −9.649022718894528597203314728578, −9.212879486606263588610389039542, −8.766752875597712008886375625666, −8.679389745117110926165653873899, −8.246795330553539931853325491137, −7.40192486737818341742223158679, −7.04444614829709831000873990967, −6.83237166736305553226023105055, −6.21493109140247131124802962384, −6.09797370327935468571856435792, −5.88960415092835783548164444614, −4.78199494654529479242810511242, −4.15497241411548340926350688589, −3.85728875618723341816388641216, −3.15630869334358871749293131531, −2.52381867671227407277573328748, −2.15700104508673027605128167644, −1.32990217166099898527693036394, −0.42027559159147366791037994264, 0.42027559159147366791037994264, 1.32990217166099898527693036394, 2.15700104508673027605128167644, 2.52381867671227407277573328748, 3.15630869334358871749293131531, 3.85728875618723341816388641216, 4.15497241411548340926350688589, 4.78199494654529479242810511242, 5.88960415092835783548164444614, 6.09797370327935468571856435792, 6.21493109140247131124802962384, 6.83237166736305553226023105055, 7.04444614829709831000873990967, 7.40192486737818341742223158679, 8.246795330553539931853325491137, 8.679389745117110926165653873899, 8.766752875597712008886375625666, 9.212879486606263588610389039542, 9.649022718894528597203314728578, 9.840462188827719874446867679209

Graph of the ZZ-function along the critical line