Properties

Label 8-2016e4-1.1-c0e4-0-7
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $1.02468$
Root an. cond. $1.00305$
Motivic weight $0$
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 + 4·23-s − 4·43-s + 4·53-s + 4·67-s − 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-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  − 16-s + 4·23-s − 4·43-s + 4·53-s + 4·67-s − 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.02468\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417570213\)
\(L(\frac12)\) \(\approx\) \(1.417570213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.75967460174509450190532711959, −6.75127767816650704362619139691, −6.46465848585577000637703961364, −6.04392121211222303345105595965, −5.74180788492969189399364843112, −5.61322515139075385520415627401, −5.47330184503695290307771097755, −5.08412844704054861597024899661, −5.03732949080677335462113143650, −4.84851355508664444668487386147, −4.73391224770654405397549027470, −4.40143779719812455335429239030, −4.01394670062281366855847775225, −3.93632007076975203731266967115, −3.54647635587387229175586828304, −3.29965736067366405116512818444, −3.27672315288221537489963663452, −2.95025200153962225945856468401, −2.50140068699164507080516132943, −2.41743102843893257105563636487, −2.11421982787502270904168949042, −1.86729051005627940736126416218, −1.20749128807090368073365284731, −1.15600757047971530270336309326, −0.69209541322626166937473202695, 0.69209541322626166937473202695, 1.15600757047971530270336309326, 1.20749128807090368073365284731, 1.86729051005627940736126416218, 2.11421982787502270904168949042, 2.41743102843893257105563636487, 2.50140068699164507080516132943, 2.95025200153962225945856468401, 3.27672315288221537489963663452, 3.29965736067366405116512818444, 3.54647635587387229175586828304, 3.93632007076975203731266967115, 4.01394670062281366855847775225, 4.40143779719812455335429239030, 4.73391224770654405397549027470, 4.84851355508664444668487386147, 5.03732949080677335462113143650, 5.08412844704054861597024899661, 5.47330184503695290307771097755, 5.61322515139075385520415627401, 5.74180788492969189399364843112, 6.04392121211222303345105595965, 6.46465848585577000637703961364, 6.75127767816650704362619139691, 6.75967460174509450190532711959

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