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Properties

Label 16-507e8-1.1-c1e8-0-3
Degree 1616
Conductor 4.366×10214.366\times 10^{21}
Sign 11
Analytic cond. 72157.372157.3
Root an. cond. 2.012062.01206
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 6·9-s + 13·16-s + 36·36-s − 6·64-s + 9·81-s + 72·121-s + 127-s + 131-s + 137-s + 139-s − 78·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3·4-s − 2·9-s + 13/4·16-s + 6·36-s − 3/4·64-s + 81-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.5·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

Λ(s)=((38⋅1316)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅1316)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 38⋅13163^{8} \cdot 13^{16}
Sign: 11
Analytic conductor: 72157.372157.3
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 38⋅1316, ( :[1/2]8), 1)(16,\ 3^{8} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.045018776810.04501877681
L(12)L(\frac12) ≈\approx 0.045018776810.04501877681
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}
ppFp(T)F_p(T)
bad3 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
13 1 1
good2 (1+3T2+7T4+3p2T6+p4T8)2 ( 1 + 3 T^{2} + 7 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} )^{2}
5 (1−2T4+p4T8)2 ( 1 - 2 T^{4} + p^{4} T^{8} )^{2}
7 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
11 (1−36T2+553T4−36p2T6+p4T8)2 ( 1 - 36 T^{2} + 553 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2}
17 (1−pT2+p2T4)4 ( 1 - p T^{2} + p^{2} T^{4} )^{4}
19 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
23 (1−pT2+p2T4)4 ( 1 - p T^{2} + p^{2} T^{4} )^{4}
29 (1+pT2+p2T4)4 ( 1 + p T^{2} + p^{2} T^{4} )^{4}
31 (1+p2T4)4 ( 1 + p^{2} T^{4} )^{4}
37 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
41 (1−36T2+2113T4−36p2T6+p4T8)2 ( 1 - 36 T^{2} + 2113 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2}
43 (1+70T2+3051T4+70p2T6+p4T8)2 ( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2}
47 (1−4370T4+p4T8)2 ( 1 - 4370 T^{4} + p^{4} T^{8} )^{2}
53 (1−pT2)8 ( 1 - p T^{2} )^{8}
59 (1−204T2+17353T4−204p2T6+p4T8)2 ( 1 - 204 T^{2} + 17353 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2}
61 (1+70T2+1179T4+70p2T6+p4T8)2 ( 1 + 70 T^{2} + 1179 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
71 (1−204T2+18913T4−204p2T6+p4T8)2 ( 1 - 204 T^{2} + 18913 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2}
73 (1+p2T4)4 ( 1 + p^{2} T^{4} )^{4}
79 (1+50T2+p2T4)4 ( 1 + 50 T^{2} + p^{2} T^{4} )^{4}
83 (1−13730T4+p4T8)2 ( 1 - 13730 T^{4} + p^{4} T^{8} )^{2}
89 (1+276T2+33313T4+276p2T6+p4T8)2 ( 1 + 276 T^{2} + 33313 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2}
97 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
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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

−4.63551374383456508573874680378, −4.52932241960333453520760884076, −4.45430129627352214445097803791, −4.42866599104787517010859463672, −4.41399924860492681751392714137, −4.38246814417981268494422555232, −3.83117910821008392603431391658, −3.71926510659350114498748403702, −3.63309509335693622623564456967, −3.58277553856310040786373012310, −3.46883369951088156917398547757, −3.33376386521510354470825606681, −2.96382707438719468846422912008, −2.82222404903425104582719977440, −2.76756259704859255798247468964, −2.46243456803328184106975842712, −2.42853375116889831060229058189, −2.21504355745170351523548726950, −1.90558844655801470196211457479, −1.69938318644832911656625623526, −1.43735324962412166409103779611, −1.03288671532338169759142507312, −0.902265000529904177501658699656, −0.43033674998247893913707154931, −0.082825186433060653568919060284, 0.082825186433060653568919060284, 0.43033674998247893913707154931, 0.902265000529904177501658699656, 1.03288671532338169759142507312, 1.43735324962412166409103779611, 1.69938318644832911656625623526, 1.90558844655801470196211457479, 2.21504355745170351523548726950, 2.42853375116889831060229058189, 2.46243456803328184106975842712, 2.76756259704859255798247468964, 2.82222404903425104582719977440, 2.96382707438719468846422912008, 3.33376386521510354470825606681, 3.46883369951088156917398547757, 3.58277553856310040786373012310, 3.63309509335693622623564456967, 3.71926510659350114498748403702, 3.83117910821008392603431391658, 4.38246814417981268494422555232, 4.41399924860492681751392714137, 4.42866599104787517010859463672, 4.45430129627352214445097803791, 4.52932241960333453520760884076, 4.63551374383456508573874680378

Graph of the ZZ-function along the critical line

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