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Properties

Label 4-132300-1.1-c1e2-0-5
Degree 44
Conductor 132300132300
Sign 11
Analytic cond. 8.435568.43556
Root an. cond. 1.704231.70423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 4·5-s + 9-s + 12-s − 4·15-s + 16-s − 4·20-s + 11·25-s − 27-s − 36-s + 4·45-s + 16·47-s − 48-s − 7·49-s + 4·60-s − 64-s − 11·75-s + 16·79-s + 4·80-s + 81-s + 24·83-s − 11·100-s + 108-s − 28·109-s − 6·121-s + 24·125-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1.78·5-s + 1/3·9-s + 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.894·20-s + 11/5·25-s − 0.192·27-s − 1/6·36-s + 0.596·45-s + 2.33·47-s − 0.144·48-s − 49-s + 0.516·60-s − 1/8·64-s − 1.27·75-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 2.63·83-s − 1.09·100-s + 0.0962·108-s − 2.68·109-s − 0.545·121-s + 2.14·125-s + ⋯

Functional equation

Λ(s)=(132300s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(132300s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 132300132300    =    22⋅33⋅52⋅722^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 8.435568.43556
Root analytic conductor: 1.704231.70423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 132300, ( :1/2,1/2), 1)(4,\ 132300,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6593970971.659397097
L(12)L(\frac12) ≈\approx 1.6593970971.659397097
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+T2 1 + T^{2}
3C1C_1 1+T 1 + T
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
7C2C_2 1+pT2 1 + p T^{2}
good11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{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

−9.382079667207611550181902865879, −9.166320354255721860880853713173, −8.527606124134460565465988984196, −7.927461294375509403356736877594, −7.36649750923666807230761191162, −6.60155933017990789780037515756, −6.42411140065541329502446951920, −5.74295706139379495360936903101, −5.38369296605186321956951583532, −4.93174302448540981311704327466, −4.27573257294083501838383539918, −3.50341666432398320311312850768, −2.59942620037353270460340022776, −1.94560952433490753418597398271, −0.988344135984823100739915057559, 0.988344135984823100739915057559, 1.94560952433490753418597398271, 2.59942620037353270460340022776, 3.50341666432398320311312850768, 4.27573257294083501838383539918, 4.93174302448540981311704327466, 5.38369296605186321956951583532, 5.74295706139379495360936903101, 6.42411140065541329502446951920, 6.60155933017990789780037515756, 7.36649750923666807230761191162, 7.927461294375509403356736877594, 8.527606124134460565465988984196, 9.166320354255721860880853713173, 9.382079667207611550181902865879

Graph of the ZZ-function along the critical line