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Properties

Label 4-600e2-1.1-c1e2-0-2
Degree 44
Conductor 360000360000
Sign 11
Analytic cond. 22.953922.9539
Root an. cond. 2.188842.18884
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 − 2·3-s + 2·4-s − 4·6-s + 3·9-s − 4·12-s − 8·13-s − 4·16-s + 6·18-s − 16·26-s − 4·27-s + 4·31-s − 8·32-s + 6·36-s − 16·37-s + 16·39-s + 4·41-s − 8·43-s + 8·48-s + 10·49-s − 16·52-s + 12·53-s − 8·54-s + 8·62-s − 8·64-s + 24·67-s + 24·71-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s − 1.15·12-s − 2.21·13-s − 16-s + 1.41·18-s − 3.13·26-s − 0.769·27-s + 0.718·31-s − 1.41·32-s + 36-s − 2.63·37-s + 2.56·39-s + 0.624·41-s − 1.21·43-s + 1.15·48-s + 10/7·49-s − 2.21·52-s + 1.64·53-s − 1.08·54-s + 1.01·62-s − 64-s + 2.93·67-s + 2.84·71-s + ⋯

Functional equation

Λ(s)=(360000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(360000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 360000360000    =    26⋅32⋅542^{6} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 22.953922.9539
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 360000, ( :1/2,1/2), 1)(4,\ 360000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6906594181.690659418
L(12)L(\frac12) ≈\approx 1.6906594181.690659418
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)
bad2C2C_2 1−pT+pT2 1 - p T + p T^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
61C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
67C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
83C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 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

−11.37331684323367922348054912130, −10.42438477100925057134788537796, −10.16937877639622175159315053319, −9.829932887489820091776894785556, −9.236559307019962860025346071624, −8.743380302330356769452213775578, −8.154389238982362704566127887242, −7.39455817417441250097264741188, −7.16239952312079613426910753772, −6.69363539304346412974409166717, −6.33381802077517724086867331514, −5.61505272819803059433681440237, −5.27213745154939131407998913986, −4.80713801216718580854087167474, −4.74817072293679982607834694681, −3.66080884828571460979894058315, −3.59109641256330707808232607961, −2.27687552175324204583534412412, −2.26658366836664787218179475693, −0.59793390921578088501730731337, 0.59793390921578088501730731337, 2.26658366836664787218179475693, 2.27687552175324204583534412412, 3.59109641256330707808232607961, 3.66080884828571460979894058315, 4.74817072293679982607834694681, 4.80713801216718580854087167474, 5.27213745154939131407998913986, 5.61505272819803059433681440237, 6.33381802077517724086867331514, 6.69363539304346412974409166717, 7.16239952312079613426910753772, 7.39455817417441250097264741188, 8.154389238982362704566127887242, 8.743380302330356769452213775578, 9.236559307019962860025346071624, 9.829932887489820091776894785556, 10.16937877639622175159315053319, 10.42438477100925057134788537796, 11.37331684323367922348054912130

Graph of the ZZ-function along the critical line