Properties

Label 4-2e16-1.1-c2e2-0-3
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  − 8·3-s + 30·9-s − 8·11-s + 36·17-s + 24·19-s + 46·25-s − 40·27-s + 64·33-s + 28·41-s + 56·43-s + 34·49-s − 288·51-s − 192·57-s − 104·59-s − 8·67-s − 132·73-s − 368·75-s − 205·81-s + 280·83-s + 60·89-s − 28·97-s − 240·99-s + 312·107-s + 196·113-s − 194·121-s − 224·123-s + 127-s + ⋯
L(s)  = 1  − 8/3·3-s + 10/3·9-s − 0.727·11-s + 2.11·17-s + 1.26·19-s + 1.83·25-s − 1.48·27-s + 1.93·33-s + 0.682·41-s + 1.30·43-s + 0.693·49-s − 5.64·51-s − 3.36·57-s − 1.76·59-s − 0.119·67-s − 1.80·73-s − 4.90·75-s − 2.53·81-s + 3.37·83-s + 0.674·89-s − 0.288·97-s − 2.42·99-s + 2.91·107-s + 1.73·113-s − 1.60·121-s − 1.82·123-s + 0.00787·127-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

\(L(\frac{3}{2})\) \(\approx\) \(0.7840484986\)
\(L(\frac12)\) \(\approx\) \(0.7840484986\)
\(L(2)\) 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$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 46 T^{2} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 34 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 142 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 542 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1486 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 898 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 1838 T^{2} + p^{4} T^{4} \)
41$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 28 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4162 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 1262 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 + 52 T + p^{2} T^{2} )^{2} \)
61$C_2^2$ \( 1 - 718 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 6946 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 66 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12226 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 140 T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
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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

−11.85582081523456332846580851048, −11.75183638602728468503981576679, −10.94818749138899270763573401898, −10.79381539689994165511275469903, −10.35484381775446605534131523715, −9.992304804043912514155658221355, −9.275914426692969173672337956050, −8.752133401977272079518951326987, −7.75578918130123040564816823938, −7.55691363291869317946501006510, −6.94962251592292609955433208616, −6.24839141963316216006437609778, −5.79713605154750991705242313590, −5.56581415042218026087079434027, −4.81287834554337716371577244410, −4.78317603164530401227627657582, −3.46868096644965774554968593071, −2.80563355901881205780746947706, −1.14443351465704640185923497858, −0.67609316544590986465374441940, 0.67609316544590986465374441940, 1.14443351465704640185923497858, 2.80563355901881205780746947706, 3.46868096644965774554968593071, 4.78317603164530401227627657582, 4.81287834554337716371577244410, 5.56581415042218026087079434027, 5.79713605154750991705242313590, 6.24839141963316216006437609778, 6.94962251592292609955433208616, 7.55691363291869317946501006510, 7.75578918130123040564816823938, 8.752133401977272079518951326987, 9.275914426692969173672337956050, 9.992304804043912514155658221355, 10.35484381775446605534131523715, 10.79381539689994165511275469903, 10.94818749138899270763573401898, 11.75183638602728468503981576679, 11.85582081523456332846580851048

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