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Properties

Label 16-960e8-1.1-c1e8-0-2
Degree 1616
Conductor 7.214×10237.214\times 10^{23}
Sign 11
Analytic cond. 1.19230×1071.19230\times 10^{7}
Root an. cond. 2.768682.76868
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  − 12·9-s + 20·25-s − 56·49-s + 90·81-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 240·225-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4·9-s + 4·25-s − 8·49-s + 10·81-s + 8/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.30·169-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 − 16·225-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

Λ(s)=((248⋅38⋅58)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((248⋅38⋅58)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 248⋅38⋅582^{48} \cdot 3^{8} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.19230×1071.19230\times 10^{7}
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 248⋅38⋅58, ( :[1/2]8), 1)(16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.17354996450.1735499645
L(12)L(\frac12) ≈\approx 0.17354996450.1735499645
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)
bad2 1 1
3 (1+pT2)4 ( 1 + p T^{2} )^{4}
5 (1−pT2)4 ( 1 - p T^{2} )^{4}
good7 (1+pT2)8 ( 1 + p T^{2} )^{8}
11 (1−2T2+p2T4)4 ( 1 - 2 T^{2} + p^{2} T^{4} )^{4}
13 (1+14T2+p2T4)4 ( 1 + 14 T^{2} + p^{2} T^{4} )^{4}
17 (1−26T2+p2T4)4 ( 1 - 26 T^{2} + p^{2} T^{4} )^{4}
19 (1+pT2)8 ( 1 + p T^{2} )^{8}
23 (1+14T2+p2T4)4 ( 1 + 14 T^{2} + p^{2} T^{4} )^{4}
29 (1+38T2+p2T4)4 ( 1 + 38 T^{2} + p^{2} T^{4} )^{4}
31 (1−58T2+p2T4)4 ( 1 - 58 T^{2} + p^{2} T^{4} )^{4}
37 (1−34T2+p2T4)4 ( 1 - 34 T^{2} + p^{2} T^{4} )^{4}
41 (1−pT2)8 ( 1 - p T^{2} )^{8}
43 (1−74T2+p2T4)4 ( 1 - 74 T^{2} + p^{2} T^{4} )^{4}
47 (1−34T2+p2T4)4 ( 1 - 34 T^{2} + p^{2} T^{4} )^{4}
53 (1−pT2)8 ( 1 - p T^{2} )^{8}
59 (1−98T2+p2T4)4 ( 1 - 98 T^{2} + p^{2} T^{4} )^{4}
61 (1−pT2)8 ( 1 - p T^{2} )^{8}
67 (1−26T2+p2T4)4 ( 1 - 26 T^{2} + p^{2} T^{4} )^{4}
71 (1+pT2)8 ( 1 + p T^{2} )^{8}
73 (1−pT2)8 ( 1 - p T^{2} )^{8}
79 (1+38T2+p2T4)4 ( 1 + 38 T^{2} + p^{2} T^{4} )^{4}
83 (1+pT2)8 ( 1 + p T^{2} )^{8}
89 (1−pT2)8 ( 1 - p T^{2} )^{8}
97 (1−pT2)8 ( 1 - p T^{2} )^{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

−4.48550660108274917741018141967, −4.36209133669922089419828175344, −3.86153309243992113464743401193, −3.74530959842003235850763108461, −3.60373583683361334787751366159, −3.59401384314558231938282281350, −3.54882392359838091800767831410, −3.30383803349316740622780350511, −3.09095662046688303152071536076, −3.08566290167545235081967977943, −2.90659460804496379766342987903, −2.88248998634005809457063568149, −2.80215764896064679445822353098, −2.63452261318683327804498077503, −2.25414915125911134041742348184, −2.18815287057687500052051311580, −2.15634412165681408961504156973, −1.97299800824828430034409551543, −1.55501838110925425058075047723, −1.29277513293041586433928372928, −1.28065355013966270050395778982, −1.20384066763146820477574700898, −0.59597037120211312759395947692, −0.46866808024713195643204059910, −0.07687759300264864242748861938, 0.07687759300264864242748861938, 0.46866808024713195643204059910, 0.59597037120211312759395947692, 1.20384066763146820477574700898, 1.28065355013966270050395778982, 1.29277513293041586433928372928, 1.55501838110925425058075047723, 1.97299800824828430034409551543, 2.15634412165681408961504156973, 2.18815287057687500052051311580, 2.25414915125911134041742348184, 2.63452261318683327804498077503, 2.80215764896064679445822353098, 2.88248998634005809457063568149, 2.90659460804496379766342987903, 3.08566290167545235081967977943, 3.09095662046688303152071536076, 3.30383803349316740622780350511, 3.54882392359838091800767831410, 3.59401384314558231938282281350, 3.60373583683361334787751366159, 3.74530959842003235850763108461, 3.86153309243992113464743401193, 4.36209133669922089419828175344, 4.48550660108274917741018141967

Graph of the ZZ-function along the critical line

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