Properties

Label 16-34e16-1.1-c0e8-0-0
Degree $16$
Conductor $3.189\times 10^{24}$
Sign $1$
Analytic cond. $0.0122721$
Root an. cond. $0.759551$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·16-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 + 3·256-s + ⋯
L(s)  = 1  − 2·16-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 + 3·256-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(0.0122721\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1156} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 17^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6945336474\)
\(L(\frac12)\) \(\approx\) \(0.6945336474\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
17 \( 1 \)
good3 \( ( 1 + T^{8} )^{2} \)
5 \( ( 1 + T^{8} )^{2} \)
7 \( ( 1 + T^{8} )^{2} \)
11 \( ( 1 + T^{8} )^{2} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
19 \( ( 1 + T^{4} )^{4} \)
23 \( ( 1 + T^{8} )^{2} \)
29 \( ( 1 + T^{8} )^{2} \)
31 \( ( 1 + T^{8} )^{2} \)
37 \( ( 1 + T^{8} )^{2} \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{4} )^{4} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 + T^{8} )^{2} \)
79 \( ( 1 + T^{8} )^{2} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 + T^{8} )^{2} \)
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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

−4.35029249334929860157248360240, −4.32105891984986622507252050702, −4.31769717812614239343102746203, −4.20793446163963183863860167720, −4.04441639423184818876657611596, −3.68121970960635462551561965975, −3.55934054757446103978096363008, −3.46921444218388980636167900408, −3.45267243443843036676194193299, −3.33991680556616519864203997718, −3.08203415893466478513500146656, −2.97284965199853163944970200054, −2.72851795655925785183081877707, −2.68459072940010737903354980328, −2.56956287288719737630987979015, −2.39286927844528915535709674927, −2.21668949643421053592745939096, −2.00114764386466823104203924190, −1.96221347260148237199134226628, −1.76431244579556931700635892351, −1.49818981190896659507726285148, −1.49805676899469575772130237740, −1.15265089224734115874199161799, −0.74323389126179422601511388332, −0.63200268437517724297310985726, 0.63200268437517724297310985726, 0.74323389126179422601511388332, 1.15265089224734115874199161799, 1.49805676899469575772130237740, 1.49818981190896659507726285148, 1.76431244579556931700635892351, 1.96221347260148237199134226628, 2.00114764386466823104203924190, 2.21668949643421053592745939096, 2.39286927844528915535709674927, 2.56956287288719737630987979015, 2.68459072940010737903354980328, 2.72851795655925785183081877707, 2.97284965199853163944970200054, 3.08203415893466478513500146656, 3.33991680556616519864203997718, 3.45267243443843036676194193299, 3.46921444218388980636167900408, 3.55934054757446103978096363008, 3.68121970960635462551561965975, 4.04441639423184818876657611596, 4.20793446163963183863860167720, 4.31769717812614239343102746203, 4.32105891984986622507252050702, 4.35029249334929860157248360240

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

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