Properties

Label 4-2e16-1.1-c2e2-0-1
Degree $4$
Conductor $65536$
Sign $1$
Analytic cond. $48.6575$
Root an. cond. $2.64111$
Motivic weight $2$
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  − 18·9-s − 60·17-s + 14·25-s − 36·41-s + 98·49-s + 220·73-s + 243·81-s + 156·89-s + 260·97-s − 60·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.08e3·153-s + 157-s + 163-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·9-s − 3.52·17-s + 0.559·25-s − 0.878·41-s + 2·49-s + 3.01·73-s + 3·81-s + 1.75·89-s + 2.68·97-s − 0.530·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 7.05·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(48.6575\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8237297921\)
\(L(\frac12)\) \(\approx\) \(0.8237297921\)
\(L(2)\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )( 1 + 8 T + p^{2} T^{2} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 120 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 78 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 130 T + p^{2} T^{2} )^{2} \)
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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

−11.84897483535717345494408360754, −11.48404658195611171149597790063, −11.17575119881244345364304208369, −10.58809857826808725540151413507, −10.47518309748217488523686237553, −9.204890837666275077978777309083, −9.194806978197714835456594616096, −8.692242625162404582461480791264, −8.346890821322585948276452864893, −7.68750611029087212451489431201, −6.91371745094398458547318174262, −6.42128316075563900156813059573, −6.18238856153069890903642972633, −5.23352131234629779474248842687, −4.93221207676176833001050767193, −4.12531440497471162362207439728, −3.43205639356193538159278500540, −2.40645241622354954004466657877, −2.28238161379059092759616284696, −0.44340363828410367266757405728, 0.44340363828410367266757405728, 2.28238161379059092759616284696, 2.40645241622354954004466657877, 3.43205639356193538159278500540, 4.12531440497471162362207439728, 4.93221207676176833001050767193, 5.23352131234629779474248842687, 6.18238856153069890903642972633, 6.42128316075563900156813059573, 6.91371745094398458547318174262, 7.68750611029087212451489431201, 8.346890821322585948276452864893, 8.692242625162404582461480791264, 9.194806978197714835456594616096, 9.204890837666275077978777309083, 10.47518309748217488523686237553, 10.58809857826808725540151413507, 11.17575119881244345364304208369, 11.48404658195611171149597790063, 11.84897483535717345494408360754

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