Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 167^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 6·12-s + 5·16-s − 6·18-s − 8·24-s + 4·27-s − 6·32-s + 9·36-s + 10·48-s + 2·49-s − 8·54-s + 7·64-s − 12·72-s + 5·81-s − 12·96-s − 4·97-s − 4·98-s + 12·108-s − 2·121-s + 127-s − 8·128-s + 131-s + ⋯
L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 6·12-s + 5·16-s − 6·18-s − 8·24-s + 4·27-s − 6·32-s + 9·36-s + 10·48-s + 2·49-s − 8·54-s + 7·64-s − 12·72-s + 5·81-s − 12·96-s − 4·97-s − 4·98-s + 12·108-s − 2·121-s + 127-s − 8·128-s + 131-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4016016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4016016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4016016\)    =    \(2^{4} \cdot 3^{2} \cdot 167^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4016016,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.201126857$
$L(\frac12)$  $\approx$  $1.201126857$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
167$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$ \( ( 1 + T )^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.214066079282701444981799988734, −9.153185246128506526850677113033, −8.879276186849607427647040622757, −8.412961246670247356577413665962, −7.987965655892556815219143801371, −7.916895219853405224120416975132, −7.41051997880905346986112716278, −7.07082369257477852149992716431, −6.67083347587613773843148875436, −6.46815055191435980337744499833, −5.59971734959201756791184949995, −5.37846181979361614360849979197, −4.36522854736731233247351960732, −4.03619821377025191371761572377, −3.41593054739066540491244199593, −3.02775246984645232156406584006, −2.47821479992055549652802069702, −2.25298910163709615521970108812, −1.52787103337297115316766718154, −1.07868558643509318116617262004, 1.07868558643509318116617262004, 1.52787103337297115316766718154, 2.25298910163709615521970108812, 2.47821479992055549652802069702, 3.02775246984645232156406584006, 3.41593054739066540491244199593, 4.03619821377025191371761572377, 4.36522854736731233247351960732, 5.37846181979361614360849979197, 5.59971734959201756791184949995, 6.46815055191435980337744499833, 6.67083347587613773843148875436, 7.07082369257477852149992716431, 7.41051997880905346986112716278, 7.916895219853405224120416975132, 7.987965655892556815219143801371, 8.412961246670247356577413665962, 8.879276186849607427647040622757, 9.153185246128506526850677113033, 9.214066079282701444981799988734

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