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Properties

Label 4-3888e2-1.1-c1e2-0-8
Degree 44
Conductor 1511654415116544
Sign 11
Analytic cond. 963.843963.843
Root an. cond. 5.571875.57187
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 10·13-s + 2·19-s + 2·25-s − 10·31-s − 2·37-s + 2·43-s − 11·49-s + 4·61-s − 16·67-s + 4·73-s + 2·79-s + 20·91-s + 34·97-s − 16·103-s + 34·109-s − 10·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.755·7-s + 2.77·13-s + 0.458·19-s + 2/5·25-s − 1.79·31-s − 0.328·37-s + 0.304·43-s − 1.57·49-s + 0.512·61-s − 1.95·67-s + 0.468·73-s + 0.225·79-s + 2.09·91-s + 3.45·97-s − 1.57·103-s + 3.25·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-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 + ⋯

Functional equation

Λ(s)=(15116544s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15116544s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1511654415116544    =    28⋅3102^{8} \cdot 3^{10}
Sign: 11
Analytic conductor: 963.843963.843
Root analytic conductor: 5.571875.57187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15116544, ( :1/2,1/2), 1)(4,\ 15116544,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6418924163.641892416
L(12)L(\frac12) ≈\approx 3.6418924163.641892416
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
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
37C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
47C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
53C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
59C22C_2^2 1+106T2+p2T4 1 + 106 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
83C22C_2^2 1+118T2+p2T4 1 + 118 T^{2} + p^{2} T^{4}
89C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
97C2C_2 (1−17T+pT2)2 ( 1 - 17 T + p 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

−8.823148884472770583686001033090, −8.292632570800360480618355268794, −7.889909888006925658409255709715, −7.72253562023634107256833095808, −7.22073864149760325444080616349, −6.81574589779543637368539971185, −6.28554038131584619415916166866, −6.19601898208036417515526512565, −5.63429678017844926628142228106, −5.42599137125184929601519370235, −4.83706904053187060669113718097, −4.58022826876496116172597702043, −3.81573367725641187139774016798, −3.80632793632654084893354210457, −3.23592837460017240011479082399, −2.92573440025065869437037121154, −1.96026140301677575807151011176, −1.72304635608116323224123903823, −1.23579631315049212244451551519, −0.59702866500578487456240907865, 0.59702866500578487456240907865, 1.23579631315049212244451551519, 1.72304635608116323224123903823, 1.96026140301677575807151011176, 2.92573440025065869437037121154, 3.23592837460017240011479082399, 3.80632793632654084893354210457, 3.81573367725641187139774016798, 4.58022826876496116172597702043, 4.83706904053187060669113718097, 5.42599137125184929601519370235, 5.63429678017844926628142228106, 6.19601898208036417515526512565, 6.28554038131584619415916166866, 6.81574589779543637368539971185, 7.22073864149760325444080616349, 7.72253562023634107256833095808, 7.889909888006925658409255709715, 8.292632570800360480618355268794, 8.823148884472770583686001033090

Graph of the ZZ-function along the critical line