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Properties

Label 4-924800-1.1-c1e2-0-8
Degree 44
Conductor 924800924800
Sign 11
Analytic cond. 58.966058.9660
Root an. cond. 2.771082.77108
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  + 6·9-s + 4·13-s + 2·17-s + 25-s − 2·49-s + 12·53-s + 27·81-s + 12·89-s − 12·101-s + 24·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·9-s + 1.10·13-s + 0.485·17-s + 1/5·25-s − 2/7·49-s + 1.64·53-s + 3·81-s + 1.27·89-s − 1.19·101-s + 2.21·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=(924800s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(924800s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 924800924800    =    27⋅52⋅1722^{7} \cdot 5^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 58.966058.9660
Root analytic conductor: 2.771082.77108
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 924800, ( :1/2,1/2), 1)(4,\ 924800,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9074093692.907409369
L(12)L(\frac12) ≈\approx 2.9074093692.907409369
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
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
17C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
83C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 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

−8.102474755417312482751636017727, −7.63480726341069678342063880921, −7.31418368250277689619278754952, −6.90489865960865564603697637591, −6.35598375216417660582625888186, −6.13116133525925096259287872344, −5.35017836730591472742155893237, −5.01749996472848131933830271333, −4.40598427015520727142670462165, −3.87831858695205618704850596461, −3.69911802761298094032213517943, −2.89831831585997321487363096349, −2.11003624829916870751595686170, −1.44917871497857968747448625423, −0.909665636801792588194732483250, 0.909665636801792588194732483250, 1.44917871497857968747448625423, 2.11003624829916870751595686170, 2.89831831585997321487363096349, 3.69911802761298094032213517943, 3.87831858695205618704850596461, 4.40598427015520727142670462165, 5.01749996472848131933830271333, 5.35017836730591472742155893237, 6.13116133525925096259287872344, 6.35598375216417660582625888186, 6.90489865960865564603697637591, 7.31418368250277689619278754952, 7.63480726341069678342063880921, 8.102474755417312482751636017727

Graph of the ZZ-function along the critical line