Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 223^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 9·11-s + 7·13-s − 4·14-s + 5·16-s + 2·17-s + 5·19-s + 18·22-s − 8·23-s − 10·25-s + 14·26-s − 6·28-s + 7·29-s + 6·31-s + 6·32-s + 4·34-s + 2·37-s + 10·38-s − 8·41-s − 3·43-s + 27·44-s − 16·46-s + 5·47-s + 2·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 2.71·11-s + 1.94·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s + 1.14·19-s + 3.83·22-s − 1.66·23-s − 2·25-s + 2.74·26-s − 1.13·28-s + 1.29·29-s + 1.07·31-s + 1.06·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s − 1.24·41-s − 0.457·43-s + 4.07·44-s − 2.35·46-s + 0.729·47-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16112196\)    =    \(2^{2} \cdot 3^{4} \cdot 223^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4014} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16112196,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $11.00033849$
$L(\frac12)$  $\approx$  $11.00033849$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;223\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;223\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
223$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 9 T + 39 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 97 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 117 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 139 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T + 77 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 206 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.452512580828103690275563107163, −8.327948808020895528790761162709, −7.75829950159078180381952489958, −7.52128082438125781523221355795, −6.78108393961842253342077941767, −6.65292165646271728631785750404, −6.29405210868210541701229249654, −6.13742942172902619036742078109, −5.66554143417133303229295296261, −5.54958076003078191977393141470, −4.54032896007714357708221393900, −4.49819863725997226427627388022, −3.85513249980711232195938008799, −3.78655078297298587025780136077, −3.30996906844965300312688807378, −3.16256737062436172329747616053, −2.24763094074381032511929667268, −1.74274490168439221789530784475, −1.29875341973171179208727592951, −0.813117710041396841667166824850, 0.813117710041396841667166824850, 1.29875341973171179208727592951, 1.74274490168439221789530784475, 2.24763094074381032511929667268, 3.16256737062436172329747616053, 3.30996906844965300312688807378, 3.78655078297298587025780136077, 3.85513249980711232195938008799, 4.49819863725997226427627388022, 4.54032896007714357708221393900, 5.54958076003078191977393141470, 5.66554143417133303229295296261, 6.13742942172902619036742078109, 6.29405210868210541701229249654, 6.65292165646271728631785750404, 6.78108393961842253342077941767, 7.52128082438125781523221355795, 7.75829950159078180381952489958, 8.327948808020895528790761162709, 8.452512580828103690275563107163

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