Properties

Label 4-24e4-1.1-c1e2-0-18
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  + 2·5-s + 2·11-s − 2·13-s + 4·17-s − 6·19-s + 2·25-s − 6·29-s + 16·31-s + 6·37-s − 10·43-s + 16·47-s + 10·49-s + 10·53-s + 4·55-s − 6·59-s − 18·61-s − 4·65-s + 10·67-s − 2·83-s + 8·85-s − 12·95-s − 4·97-s − 22·101-s − 14·107-s + 6·109-s + 12·113-s + 2·121-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 2/5·25-s − 1.11·29-s + 2.87·31-s + 0.986·37-s − 1.52·43-s + 2.33·47-s + 10/7·49-s + 1.37·53-s + 0.539·55-s − 0.781·59-s − 2.30·61-s − 0.496·65-s + 1.22·67-s − 0.219·83-s + 0.867·85-s − 1.23·95-s − 0.406·97-s − 2.18·101-s − 1.35·107-s + 0.574·109-s + 1.12·113-s + 2/11·121-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

\(L(1)\) \(\approx\) \(2.236819119\)
\(L(\frac12)\) \(\approx\) \(2.236819119\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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

−10.77311636803085939777919533946, −10.49457362522048707912170185274, −9.985492334308252459548449454653, −9.738002026339452232707527063661, −9.238224148555721332799289097895, −8.827648173254806009050575758447, −8.291743091132158793423402148583, −7.952217381380142089495914242612, −7.26456772043254885247206773098, −6.92280067119968710436887352894, −6.26462668120333481933310169658, −6.01316137671141209065033266010, −5.52372752609787966568192207742, −4.90345101020265287924670550326, −4.27159379593098255531242198801, −3.96234967114717235772572934758, −2.94986240719238815556388368069, −2.53296449977388423632414205611, −1.79537193214410255519390090922, −0.896756672785785301727100718539, 0.896756672785785301727100718539, 1.79537193214410255519390090922, 2.53296449977388423632414205611, 2.94986240719238815556388368069, 3.96234967114717235772572934758, 4.27159379593098255531242198801, 4.90345101020265287924670550326, 5.52372752609787966568192207742, 6.01316137671141209065033266010, 6.26462668120333481933310169658, 6.92280067119968710436887352894, 7.26456772043254885247206773098, 7.952217381380142089495914242612, 8.291743091132158793423402148583, 8.827648173254806009050575758447, 9.238224148555721332799289097895, 9.738002026339452232707527063661, 9.985492334308252459548449454653, 10.49457362522048707912170185274, 10.77311636803085939777919533946

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