Properties

Label 4-2e16-1.1-c5e2-0-8
Degree $4$
Conductor $65536$
Sign $1$
Analytic cond. $1685.78$
Root an. cond. $6.40767$
Motivic weight $5$
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  + 176·7-s + 342·9-s + 1.18e3·17-s + 8.20e3·23-s + 3.33e3·25-s + 8.51e3·31-s − 3.44e4·41-s − 2.59e3·47-s − 1.03e4·49-s + 6.01e4·63-s + 9.37e4·71-s − 1.35e5·73-s − 1.53e5·79-s + 5.79e4·81-s − 5.95e4·89-s − 2.44e5·97-s + 5.45e4·103-s − 5.92e4·113-s + 2.09e5·119-s + 3.05e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.06e5·153-s + ⋯
L(s)  = 1  + 1.35·7-s + 1.40·9-s + 0.996·17-s + 3.23·23-s + 1.06·25-s + 1.59·31-s − 3.20·41-s − 0.171·47-s − 0.617·49-s + 1.91·63-s + 2.20·71-s − 2.96·73-s − 2.77·79-s + 0.980·81-s − 0.796·89-s − 2.64·97-s + 0.506·103-s − 0.436·113-s + 1.35·119-s + 0.189·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 1.40·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(1685.78\)
Root analytic conductor: \(6.40767\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{256} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.483151665\)
\(L(\frac12)\) \(\approx\) \(5.483151665\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 38 p^{2} T^{2} + p^{10} T^{4} \)
5$C_2^2$ \( 1 - 3334 T^{2} + p^{10} T^{4} \)
7$C_2$ \( ( 1 - 88 T + p^{5} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 30502 T^{2} + p^{10} T^{4} \)
13$C_2^2$ \( 1 - 567862 T^{2} + p^{10} T^{4} \)
17$C_2$ \( ( 1 - 594 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 11782 p^{2} T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 - 4104 T + p^{5} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 40669462 T^{2} + p^{10} T^{4} \)
31$C_2$ \( ( 1 - 4256 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 138599110 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 17226 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 147606886 T^{2} + p^{10} T^{4} \)
47$C_2$ \( ( 1 + 1296 T + p^{5} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 456374950 T^{2} + p^{10} T^{4} \)
59$C_2^2$ \( 1 - 1371050374 T^{2} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - 482463958 T^{2} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 2224486870 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 46872 T + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 67562 T + p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 76912 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 3292624630 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 29754 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 122398 T + p^{5} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30640195795238470713041236165, −10.96673501532699991965900545132, −10.49512921935053364460954883036, −9.800847885907381303845183186686, −9.800052300470994582882185240025, −8.801830771852822067513318981193, −8.463440606133406618673717390408, −8.116790953614393569051454657527, −7.31579272115181977647119822190, −6.85982653732257161847051811756, −6.81825283350181482479339067061, −5.63731839674385602910431264745, −5.07680719835353065028031038376, −4.71642954700158013437737952717, −4.34721650962262878894440722553, −3.16123620635294066555346658567, −2.99849158018346843442178430617, −1.58855452875001172237472826631, −1.42841803377364697087145988828, −0.73608196413052327172784557268, 0.73608196413052327172784557268, 1.42841803377364697087145988828, 1.58855452875001172237472826631, 2.99849158018346843442178430617, 3.16123620635294066555346658567, 4.34721650962262878894440722553, 4.71642954700158013437737952717, 5.07680719835353065028031038376, 5.63731839674385602910431264745, 6.81825283350181482479339067061, 6.85982653732257161847051811756, 7.31579272115181977647119822190, 8.116790953614393569051454657527, 8.463440606133406618673717390408, 8.801830771852822067513318981193, 9.800052300470994582882185240025, 9.800847885907381303845183186686, 10.49512921935053364460954883036, 10.96673501532699991965900545132, 11.30640195795238470713041236165

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