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Properties

Label 16-2664e8-1.1-c0e8-0-1
Degree 1616
Conductor 2.537×10272.537\times 10^{27}
Sign 11
Analytic cond. 9.761819.76181
Root an. cond. 1.153041.15304
Motivic weight 00
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  − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + ⋯
L(s)  = 1  − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + ⋯

Functional equation

Λ(s)=((224⋅316⋅378)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((224⋅316⋅378)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 224⋅316⋅3782^{24} \cdot 3^{16} \cdot 37^{8}
Sign: 11
Analytic conductor: 9.761819.76181
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 224⋅316⋅378, ( :[0]8), 1)(16,\ 2^{24} \cdot 3^{16} \cdot 37^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2151125741.215112574
L(12)L(\frac12) ≈\approx 1.2151125741.215112574
L(1)L(1) 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}
ppFp(T)F_p(T)
bad2 1+T8 1 + T^{8}
3 1 1
37 (1+T2)4 ( 1 + T^{2} )^{4}
good5 (1+T8)2 ( 1 + T^{8} )^{2}
7 (1+T4)4 ( 1 + T^{4} )^{4}
11 (1+T2)8 ( 1 + T^{2} )^{8}
13 (1+T2)8 ( 1 + T^{2} )^{8}
17 (1+T8)2 ( 1 + T^{8} )^{2}
19 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
23 (1+T8)2 ( 1 + T^{8} )^{2}
29 (1+T8)2 ( 1 + T^{8} )^{2}
31 (1+T2)8 ( 1 + T^{2} )^{8}
41 (1+T2)8 ( 1 + T^{2} )^{8}
43 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
47 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
53 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
59 (1+T8)2 ( 1 + T^{8} )^{2}
61 (1+T2)8 ( 1 + T^{2} )^{8}
67 (1+T4)4 ( 1 + T^{4} )^{4}
71 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
73 (1+T4)4 ( 1 + T^{4} )^{4}
79 (1+T2)8 ( 1 + T^{2} )^{8}
83 (1+T2)8 ( 1 + T^{2} )^{8}
89 (1+T8)2 ( 1 + T^{8} )^{2}
97 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.80449999953140898294236880236, −3.79818255645462344592512164493, −3.73357428342909657581953319407, −3.72433079740208203324357275339, −3.64259185657015584920750699319, −3.24879307978255272710451852445, −3.04286150509872071030676768868, −3.03086757903055465057292838216, −2.95350266444204681305656460230, −2.79087697507718456370747144481, −2.59515814488071465684412498713, −2.59298913796615371330345718112, −2.58386153475641813255657178739, −2.50803906633561769262119427533, −2.16275827221005402632779570717, −1.97360262838182725643267227920, −1.79208364576888426890910647434, −1.60554284171153691975487844408, −1.59495025831135935076569858307, −1.55610531745039949362642044634, −1.48576644745438471362919550044, −0.959552126782797026366002181030, −0.874008472681171131240413780436, −0.795240026356658582282098779273, −0.35051348728392832652617393472, 0.35051348728392832652617393472, 0.795240026356658582282098779273, 0.874008472681171131240413780436, 0.959552126782797026366002181030, 1.48576644745438471362919550044, 1.55610531745039949362642044634, 1.59495025831135935076569858307, 1.60554284171153691975487844408, 1.79208364576888426890910647434, 1.97360262838182725643267227920, 2.16275827221005402632779570717, 2.50803906633561769262119427533, 2.58386153475641813255657178739, 2.59298913796615371330345718112, 2.59515814488071465684412498713, 2.79087697507718456370747144481, 2.95350266444204681305656460230, 3.03086757903055465057292838216, 3.04286150509872071030676768868, 3.24879307978255272710451852445, 3.64259185657015584920750699319, 3.72433079740208203324357275339, 3.73357428342909657581953319407, 3.79818255645462344592512164493, 3.80449999953140898294236880236

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.