Properties

Label 16-42e16-1.1-c1e8-0-1
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $1.54954\times 10^{9}$
Root an. cond. $3.75308$
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  − 32·25-s + 32·37-s + 32·43-s + 32·67-s − 32·109-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 6.39·25-s + 5.26·37-s + 4.87·43-s + 3.90·67-s − 3.06·109-s + 6.54·121-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 + 4.92·169-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 + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.54954\times 10^{9}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.461960751\)
\(L(\frac12)\) \(\approx\) \(1.461960751\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 16 T^{2} + 112 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 32 T^{2} + 672 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 60 T^{2} + 1590 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 68 T^{2} + 2326 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 16 T^{2} - 992 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - 20 T^{2} + 2950 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 48 T^{2} + 1586 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 36 T^{2} + 5718 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 80 T^{2} + 4240 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 20 T^{2} + 8134 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 144 T^{2} + 11424 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 204 T^{2} + 22134 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 144 T^{2} + 10368 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 240 T^{2} + 28800 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.13648333720564468246381547062, −3.78578215411979472901140963269, −3.60985095917530594568726609365, −3.56488869578384213870549419969, −3.51692430702604402084231443090, −3.37279743824186146531828830380, −3.21259459441733843001469707578, −2.98135514477969966478005719540, −2.83347501160713051300221448807, −2.83002698519246881588307245420, −2.46605597469200169521236291768, −2.35783055219330457216691475506, −2.30358030996254581886363231025, −2.23331809833142850814003886467, −2.16610652472045546792317432576, −2.11951525042794742093320301519, −1.84759978399482517589049708065, −1.54881324916276920801655212007, −1.53238868421994940517219302384, −1.23170150080198951957710138075, −0.949516211440669714084407610682, −0.76705830059490482459252275948, −0.67520443543456329616929782731, −0.62458896940366802826227240750, −0.10506783305890850762747595503, 0.10506783305890850762747595503, 0.62458896940366802826227240750, 0.67520443543456329616929782731, 0.76705830059490482459252275948, 0.949516211440669714084407610682, 1.23170150080198951957710138075, 1.53238868421994940517219302384, 1.54881324916276920801655212007, 1.84759978399482517589049708065, 2.11951525042794742093320301519, 2.16610652472045546792317432576, 2.23331809833142850814003886467, 2.30358030996254581886363231025, 2.35783055219330457216691475506, 2.46605597469200169521236291768, 2.83002698519246881588307245420, 2.83347501160713051300221448807, 2.98135514477969966478005719540, 3.21259459441733843001469707578, 3.37279743824186146531828830380, 3.51692430702604402084231443090, 3.56488869578384213870549419969, 3.60985095917530594568726609365, 3.78578215411979472901140963269, 4.13648333720564468246381547062

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

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