Properties

Label 4-2e16-1.1-c5e2-0-2
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  − 48·7-s + 86·9-s − 2.39e3·17-s + 368·23-s + 774·25-s + 1.14e4·31-s + 1.77e4·41-s − 4.73e4·47-s − 3.18e4·49-s − 4.12e3·63-s − 6.39e4·71-s + 9.77e3·73-s − 8.91e4·79-s − 5.16e4·81-s − 1.43e5·89-s + 9.77e4·97-s + 3.60e5·103-s − 4.70e4·113-s + 1.15e5·119-s + 3.06e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.06e5·153-s + ⋯
L(s)  = 1  − 0.370·7-s + 0.353·9-s − 2.01·17-s + 0.145·23-s + 0.247·25-s + 2.14·31-s + 1.65·41-s − 3.12·47-s − 1.89·49-s − 0.131·63-s − 1.50·71-s + 0.214·73-s − 1.60·79-s − 0.874·81-s − 1.92·89-s + 1.05·97-s + 3.35·103-s − 0.346·113-s + 0.744·119-s + 1.90·121-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 − 0.711·153-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.194054253\)
\(L(\frac12)\) \(\approx\) \(1.194054253\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 - 86 T^{2} + p^{10} T^{4} \)
5$C_2^2$ \( 1 - 774 T^{2} + p^{10} T^{4} \)
7$C_2$ \( ( 1 + 24 T + p^{5} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 306726 T^{2} + p^{10} T^{4} \)
13$C_2^2$ \( 1 - 514102 T^{2} + p^{10} T^{4} \)
17$C_2$ \( ( 1 + 1198 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 4313738 T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 - 8 p T + p^{5} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 30250774 T^{2} + p^{10} T^{4} \)
31$C_2$ \( ( 1 - 5728 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 32061638 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 8886 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 209597542 T^{2} + p^{10} T^{4} \)
47$C_2$ \( ( 1 + 23664 T + p^{5} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 699828390 T^{2} + p^{10} T^{4} \)
59$C_2^2$ \( 1 - 1145049222 T^{2} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - 1347608278 T^{2} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 2459007190 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 31960 T + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4886 T + p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 44560 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 3340172790 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 71994 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 48866 T + p^{5} T^{2} )^{2} \)
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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

−11.40307937028845557837432240109, −11.11978352593331766964150968316, −10.32474364038249819197637538675, −9.886029529969304667063226540731, −9.685606024860844524266183740564, −8.748820503442784800962148963092, −8.719924864879738623924814177993, −8.016298864661784869699502929620, −7.42771071193203155865829207830, −6.84726858017668134113999085001, −6.28000939179846108232812767170, −6.19369643979329962808526480953, −5.10600479756497898249165081885, −4.47251497737943694028708211884, −4.37991032047716966787286406760, −3.26016466579534332080095697165, −2.81664970818251678839883606371, −2.00778477468551833986573813147, −1.27978740445730790762136059629, −0.31672458671678899582905607413, 0.31672458671678899582905607413, 1.27978740445730790762136059629, 2.00778477468551833986573813147, 2.81664970818251678839883606371, 3.26016466579534332080095697165, 4.37991032047716966787286406760, 4.47251497737943694028708211884, 5.10600479756497898249165081885, 6.19369643979329962808526480953, 6.28000939179846108232812767170, 6.84726858017668134113999085001, 7.42771071193203155865829207830, 8.016298864661784869699502929620, 8.719924864879738623924814177993, 8.748820503442784800962148963092, 9.685606024860844524266183740564, 9.886029529969304667063226540731, 10.32474364038249819197637538675, 11.11978352593331766964150968316, 11.40307937028845557837432240109

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