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Properties

Label 4-855e2-1.1-c1e2-0-11
Degree 44
Conductor 731025731025
Sign −1-1
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 8·13-s − 3·16-s + 4·19-s − 25-s + 8·37-s + 8·43-s + 2·49-s − 8·52-s + 4·61-s − 7·64-s − 16·67-s + 4·73-s + 4·76-s − 8·97-s − 100-s − 32·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/2·4-s − 2.21·13-s − 3/4·16-s + 0.917·19-s − 1/5·25-s + 1.31·37-s + 1.21·43-s + 2/7·49-s − 1.10·52-s + 0.512·61-s − 7/8·64-s − 1.95·67-s + 0.468·73-s + 0.458·76-s − 0.812·97-s − 0.0999·100-s − 3.06·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 731025, ( :1/2,1/2), −1)(4,\ 731025,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5C2C_2 1+T2 1 + T^{2}
19C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
23C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
47C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
53C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2×\timesC2C_2 (1+pT2)(1+16T+pT2) ( 1 + p T^{2} )( 1 + 16 T + p T^{2} )
71C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
89C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 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

−7.88325852680582646292256615361, −7.57863096296953310313537947065, −7.16786350306497184470487891244, −6.94671668628964351262283859265, −6.26804644643691712661870554994, −5.81991575287185315972703839374, −5.32177139762261023814575475266, −4.77833801709248731889717663410, −4.44859857245519312909598910455, −3.83770466673913712729436913482, −2.99254449026549086361235189107, −2.55716925504772308443173029215, −2.19853647454185692756516131377, −1.19705306068655836886154729814, 0, 1.19705306068655836886154729814, 2.19853647454185692756516131377, 2.55716925504772308443173029215, 2.99254449026549086361235189107, 3.83770466673913712729436913482, 4.44859857245519312909598910455, 4.77833801709248731889717663410, 5.32177139762261023814575475266, 5.81991575287185315972703839374, 6.26804644643691712661870554994, 6.94671668628964351262283859265, 7.16786350306497184470487891244, 7.57863096296953310313537947065, 7.88325852680582646292256615361

Graph of the ZZ-function along the critical line