Properties

Label 16-1575e8-1.1-c1e8-0-9
Degree $16$
Conductor $3.787\times 10^{25}$
Sign $1$
Analytic cond. $6.25837\times 10^{8}$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16-s + 28·49-s + 16·67-s − 32·79-s + ⋯
L(s)  = 1  + 1/4·16-s + 4·49-s + 1.95·67-s − 3.60·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.25837\times 10^{8}\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.800621252\)
\(L(\frac12)\) \(\approx\) \(6.800621252\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( ( 1 - p T^{2} )^{4} \)
good2 \( 1 - T^{4} - 15 T^{8} - p^{4} T^{12} + p^{8} T^{16} \)
11 \( 1 + 206 T^{4} + 27795 T^{8} + 206 p^{4} T^{12} + p^{8} T^{16} \)
13 \( ( 1 - p T^{2} )^{8} \)
17 \( ( 1 + p T^{2} )^{8} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( 1 + 734 T^{4} + 258915 T^{8} + 734 p^{4} T^{12} + p^{8} T^{16} \)
29 \( 1 - 1234 T^{4} + 815475 T^{8} - 1234 p^{4} T^{12} + p^{8} T^{16} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + p T^{2} )^{8} \)
43 \( ( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 - p T^{2} )^{8} \)
67 \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( 1 - 2914 T^{4} - 16920285 T^{8} - 2914 p^{4} T^{12} + p^{8} T^{16} \)
73 \( ( 1 - p T^{2} )^{8} \)
79 \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + p T^{2} )^{8} \)
89 \( ( 1 + p T^{2} )^{8} \)
97 \( ( 1 - p T^{2} )^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.02355031197855356621449504508, −3.94512012018798983041800085711, −3.91144212844705114960165969632, −3.52441346647648835104334393401, −3.35123965533430237984100004821, −3.30700524396969132707428119629, −3.14847183620569268194505576496, −3.14155093669440759857685838918, −3.01627005852029714702930998479, −2.90469805584376748263793799378, −2.79275729467356571443313324340, −2.57082527766247969257001995481, −2.24426265192006750959374097824, −2.16193280992921388330060490960, −2.14026220681140735963012267222, −2.01612658570892580685480074182, −1.92599488685908107481609489566, −1.61128340345755791261263927027, −1.56029289479008774501049981457, −1.31104758796208904439902892911, −0.991840173411988390775062656006, −0.71133568954595936801040712235, −0.70615572658090336274763931180, −0.68635862644208209705479853733, −0.24084251852495521564969714306, 0.24084251852495521564969714306, 0.68635862644208209705479853733, 0.70615572658090336274763931180, 0.71133568954595936801040712235, 0.991840173411988390775062656006, 1.31104758796208904439902892911, 1.56029289479008774501049981457, 1.61128340345755791261263927027, 1.92599488685908107481609489566, 2.01612658570892580685480074182, 2.14026220681140735963012267222, 2.16193280992921388330060490960, 2.24426265192006750959374097824, 2.57082527766247969257001995481, 2.79275729467356571443313324340, 2.90469805584376748263793799378, 3.01627005852029714702930998479, 3.14155093669440759857685838918, 3.14847183620569268194505576496, 3.30700524396969132707428119629, 3.35123965533430237984100004821, 3.52441346647648835104334393401, 3.91144212844705114960165969632, 3.94512012018798983041800085711, 4.02355031197855356621449504508

Graph of the $Z$-function along the critical line

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