Properties

Label 4-2e16-1.1-c5e2-0-0
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

\(L(3)\) \(\approx\) \(0.1870435552\)
\(L(\frac12)\) \(\approx\) \(0.1870435552\)
\(L(\frac{7}{2})\) 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

−11.60541226325730901937735502807, −10.55201879853308613260439031575, −10.42232115203152666088650875087, −9.869342503231410947193766446648, −9.752167845702409155893436600803, −9.126851443848936814166943575674, −8.444285807094076060195889002243, −7.939785486910255471890727563564, −7.42877401219022593199367331278, −6.81986702064647802816185123443, −6.54594806974167425351621693991, −5.85997406064744443370246341093, −5.38101408225905076987449756363, −4.56262939147201569380288848886, −3.92466635453073841614723316181, −3.51872166651258097621825353372, −2.88284002834170367663423348140, −1.76578556463569406586695507691, −1.47415393542951126521269706345, −0.11909709289723749298048491877, 0.11909709289723749298048491877, 1.47415393542951126521269706345, 1.76578556463569406586695507691, 2.88284002834170367663423348140, 3.51872166651258097621825353372, 3.92466635453073841614723316181, 4.56262939147201569380288848886, 5.38101408225905076987449756363, 5.85997406064744443370246341093, 6.54594806974167425351621693991, 6.81986702064647802816185123443, 7.42877401219022593199367331278, 7.939785486910255471890727563564, 8.444285807094076060195889002243, 9.126851443848936814166943575674, 9.752167845702409155893436600803, 9.869342503231410947193766446648, 10.42232115203152666088650875087, 10.55201879853308613260439031575, 11.60541226325730901937735502807

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