Properties

Label 4-24e4-1.1-c1e2-0-26
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 12·13-s + 6·25-s + 4·37-s − 14·49-s + 20·61-s + 12·73-s − 36·97-s + 12·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.32·13-s + 6/5·25-s + 0.657·37-s − 2·49-s + 2.56·61-s + 1.40·73-s − 3.65·97-s + 1.14·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.291728606\)
\(L(\frac12)\) \(\approx\) \(2.291728606\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.612772853079119873302234581371, −8.333949469385387295662212740757, −8.142751074316513789502919252186, −7.33244289349889698794346569795, −6.76934202435783070282666272762, −6.25541731904712913819044308910, −6.20341169361318223940088511214, −5.37375392831683105890386008983, −5.02525715711667990043884728260, −4.11036595013942989412602319905, −3.81076717999348126429094994331, −3.29164018328411348390302871759, −2.57992526666733709076038094728, −1.52256003301293920411212617380, −1.00089337770198902241003598585, 1.00089337770198902241003598585, 1.52256003301293920411212617380, 2.57992526666733709076038094728, 3.29164018328411348390302871759, 3.81076717999348126429094994331, 4.11036595013942989412602319905, 5.02525715711667990043884728260, 5.37375392831683105890386008983, 6.20341169361318223940088511214, 6.25541731904712913819044308910, 6.76934202435783070282666272762, 7.33244289349889698794346569795, 8.142751074316513789502919252186, 8.333949469385387295662212740757, 8.612772853079119873302234581371

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