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Properties

Label 4-728e2-1.1-c1e2-0-16
Degree 44
Conductor 529984529984
Sign 11
Analytic cond. 33.792233.7922
Root an. cond. 2.411032.41103
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  + 2·3-s + 4·7-s + 9-s − 12·19-s + 8·21-s + 6·25-s + 2·27-s + 4·29-s − 12·31-s + 16·37-s + 20·47-s + 9·49-s + 4·53-s − 24·57-s + 8·59-s + 4·63-s + 12·75-s + 4·81-s + 12·83-s + 8·87-s − 24·93-s − 12·103-s − 8·109-s + 32·111-s + 14·113-s − 15·121-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s − 2.75·19-s + 1.74·21-s + 6/5·25-s + 0.384·27-s + 0.742·29-s − 2.15·31-s + 2.63·37-s + 2.91·47-s + 9/7·49-s + 0.549·53-s − 3.17·57-s + 1.04·59-s + 0.503·63-s + 1.38·75-s + 4/9·81-s + 1.31·83-s + 0.857·87-s − 2.48·93-s − 1.18·103-s − 0.766·109-s + 3.03·111-s + 1.31·113-s − 1.36·121-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=(529984s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(529984s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 529984529984    =    26⋅72⋅1322^{6} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 33.792233.7922
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 529984, ( :1/2,1/2), 1)(4,\ 529984,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4250949783.425094978
L(12)L(\frac12) ≈\approx 3.4250949783.425094978
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
7C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
13C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2×\timesC2C_2 (1−pT+pT2)(1+T+pT2) ( 1 - p T + p T^{2} )( 1 + T + p T^{2} )
5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
17C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C22C_2^2 1−15T2+p2T4 1 - 15 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1+5T+pT2)(1+7T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−9T+pT2)(1−7T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} )
41C22C_2^2 1+63T2+p2T4 1 + 63 T^{2} + p^{2} T^{4}
43C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−11T+pT2)(1−9T+pT2) ( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
61C22C_2^2 1−31T2+p2T4 1 - 31 T^{2} + p^{2} T^{4}
67C22C_2^2 1+71T2+p2T4 1 + 71 T^{2} + p^{2} T^{4}
71C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
73C22C_2^2 1+55T2+p2T4 1 + 55 T^{2} + p^{2} T^{4}
79C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89C22C_2^2 1−110T2+p2T4 1 - 110 T^{2} + p^{2} T^{4}
97C22C_2^2 1−41T2+p2T4 1 - 41 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

−8.591237776065458957815717100656, −8.186396113827957594491553827864, −7.64328335999930451686515524910, −7.35293630424942421864765130650, −6.73032923751594741389024075974, −6.23262449838928905777935770488, −5.69731013514640098640396991738, −5.08723474259735470485098049520, −4.57757998992851307994405067926, −4.07277619574273699897386794950, −3.82927678109542374848938982943, −2.65023581762938548003097866108, −2.50346250579294179784168117969, −1.90392920847006658130137912002, −0.940108646971592877119260915250, 0.940108646971592877119260915250, 1.90392920847006658130137912002, 2.50346250579294179784168117969, 2.65023581762938548003097866108, 3.82927678109542374848938982943, 4.07277619574273699897386794950, 4.57757998992851307994405067926, 5.08723474259735470485098049520, 5.69731013514640098640396991738, 6.23262449838928905777935770488, 6.73032923751594741389024075974, 7.35293630424942421864765130650, 7.64328335999930451686515524910, 8.186396113827957594491553827864, 8.591237776065458957815717100656

Graph of the ZZ-function along the critical line