Properties

Label 4-684e2-1.1-c2e2-0-0
Degree $4$
Conductor $467856$
Sign $1$
Analytic cond. $347.362$
Root an. cond. $4.31713$
Motivic weight $2$
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  + 6·5-s − 10·7-s − 33·13-s + 6·17-s − 26·19-s − 24·23-s + 25·25-s − 54·29-s − 60·35-s + 72·41-s + 25·43-s − 42·47-s − 23·49-s − 108·53-s − 126·59-s − 43·61-s − 198·65-s + 99·67-s − 108·71-s − 11·73-s + 3·79-s − 252·83-s + 36·85-s + 18·89-s + 330·91-s − 156·95-s − 228·97-s + ⋯
L(s)  = 1  + 6/5·5-s − 1.42·7-s − 2.53·13-s + 6/17·17-s − 1.36·19-s − 1.04·23-s + 25-s − 1.86·29-s − 1.71·35-s + 1.75·41-s + 0.581·43-s − 0.893·47-s − 0.469·49-s − 2.03·53-s − 2.13·59-s − 0.704·61-s − 3.04·65-s + 1.47·67-s − 1.52·71-s − 0.150·73-s + 3/79·79-s − 3.03·83-s + 0.423·85-s + 0.202·89-s + 3.62·91-s − 1.64·95-s − 2.35·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(467856\)    =    \(2^{4} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(347.362\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{684} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 467856,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06464347841\)
\(L(\frac12)\) \(\approx\) \(0.06464347841\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 + 26 T + p^{2} T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 11 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 33 T + 532 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 6 T - 253 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 24 T + 47 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 54 T + 1813 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 1055 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 + 937 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 72 T + 3409 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 25 T - 1224 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 42 T - 445 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2^2$ \( 1 + 108 T + 6697 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 126 T + 8773 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 43 T - 1872 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 99 T + 7756 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2^2$ \( 1 + 108 T + 8929 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \)
73$C_2^2$ \( 1 + 11 T - 5208 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 3 T + 6244 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 126 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 8029 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 228 T + 26737 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} \)
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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

−10.56647384704009064982594692848, −9.881624878995777175377063863104, −9.579117491332307959902552706799, −9.367029836696207122995430480782, −9.197994054782095037874489469637, −8.233802395712900664747308924230, −7.88233313358470050031356328436, −7.28543250336424490121613075555, −7.04105015892287891270121439627, −6.28374200190273088852360084825, −6.15051146277458375813302244145, −5.67325995674424755030461103674, −5.09495717838486606591370473522, −4.48749437261910730339440740044, −4.11929292337683045969788036633, −3.00913667717193496210934945475, −2.94795065271375083421373374208, −2.08630420622431793551764499965, −1.71518724832366238203402832107, −0.082841613900554177401101565810, 0.082841613900554177401101565810, 1.71518724832366238203402832107, 2.08630420622431793551764499965, 2.94795065271375083421373374208, 3.00913667717193496210934945475, 4.11929292337683045969788036633, 4.48749437261910730339440740044, 5.09495717838486606591370473522, 5.67325995674424755030461103674, 6.15051146277458375813302244145, 6.28374200190273088852360084825, 7.04105015892287891270121439627, 7.28543250336424490121613075555, 7.88233313358470050031356328436, 8.233802395712900664747308924230, 9.197994054782095037874489469637, 9.367029836696207122995430480782, 9.579117491332307959902552706799, 9.881624878995777175377063863104, 10.56647384704009064982594692848

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