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Properties

Label 4-1134e2-1.1-c1e2-0-45
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 5·7-s − 4·8-s + 4·13-s − 10·14-s + 5·16-s + 6·17-s − 2·19-s − 3·23-s + 5·25-s − 8·26-s + 15·28-s − 6·29-s + 10·31-s − 6·32-s − 12·34-s − 8·37-s + 4·38-s + 3·41-s − 2·43-s + 6·46-s + 6·47-s + 18·49-s − 10·50-s + 12·52-s + 6·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.88·7-s − 1.41·8-s + 1.10·13-s − 2.67·14-s + 5/4·16-s + 1.45·17-s − 0.458·19-s − 0.625·23-s + 25-s − 1.56·26-s + 2.83·28-s − 1.11·29-s + 1.79·31-s − 1.06·32-s − 2.05·34-s − 1.31·37-s + 0.648·38-s + 0.468·41-s − 0.304·43-s + 0.884·46-s + 0.875·47-s + 18/7·49-s − 1.41·50-s + 1.66·52-s + 0.824·53-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 1.8074239231.807423923
L(12)L(\frac12) ≈\approx 1.8074239231.807423923
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−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C22C_2^2 1−4T+3T2−4pT3+p2T4 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+3T−14T2+3pT3+p2T4 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+6T+7T2+6pT3+p2T4 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4}
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
37C22C_2^2 1+8T+27T2+8pT3+p2T4 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−3T−32T2−3pT3+p2T4 1 - 3 T - 32 T^{2} - 3 p T^{3} + 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 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C2C_2 (1−15T+pT2)2 ( 1 - 15 T + p T^{2} )^{2}
73C22C_2^2 1+11T+48T2+11pT3+p2T4 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4}
79C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
83C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
89C22C_2^2 1−9T−8T2−9pT3+p2T4 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+2T−93T2+2pT3+p2T4 1 + 2 T - 93 T^{2} + 2 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

−10.00762452544365622256816864013, −9.646826547508947126106558904146, −9.084912266370969053508678841913, −8.629019942972356023061823142025, −8.349843226609282130415069876291, −8.218229652425791815592869543154, −7.60498116260930810576729796684, −7.45400219865103127652679304904, −6.80832067136748689429010235194, −6.40811020488025342340455604687, −5.69775137207133593565392116214, −5.61715880863236205190427037923, −4.84657227239813815111995490214, −4.51186295546731558450092998557, −3.56212991025573369338545002227, −3.46058886128328040526502467195, −2.28677352455988310763422505300, −2.09709791278411172227671144391, −1.18571287118490488614326846276, −0.919365518272345084312670568734, 0.919365518272345084312670568734, 1.18571287118490488614326846276, 2.09709791278411172227671144391, 2.28677352455988310763422505300, 3.46058886128328040526502467195, 3.56212991025573369338545002227, 4.51186295546731558450092998557, 4.84657227239813815111995490214, 5.61715880863236205190427037923, 5.69775137207133593565392116214, 6.40811020488025342340455604687, 6.80832067136748689429010235194, 7.45400219865103127652679304904, 7.60498116260930810576729796684, 8.218229652425791815592869543154, 8.349843226609282130415069876291, 8.629019942972356023061823142025, 9.084912266370969053508678841913, 9.646826547508947126106558904146, 10.00762452544365622256816864013

Graph of the ZZ-function along the critical line