Properties

Degree 4
Conductor $ 3^{2} \cdot 23^{2} \cdot 29^{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 + 2·4-s − 4·6-s − 2·8-s + 3·9-s + 4·12-s + 3·16-s − 6·18-s − 4·24-s + 2·25-s + 4·27-s − 2·31-s − 4·32-s + 6·36-s + 2·41-s − 2·47-s + 6·48-s + 2·49-s − 4·50-s − 8·54-s + 4·62-s + 4·64-s − 6·72-s − 2·73-s + 4·75-s + 5·81-s + ⋯
L(s)  = 1  − 2·2-s + 2·3-s + 2·4-s − 4·6-s − 2·8-s + 3·9-s + 4·12-s + 3·16-s − 6·18-s − 4·24-s + 2·25-s + 4·27-s − 2·31-s − 4·32-s + 6·36-s + 2·41-s − 2·47-s + 6·48-s + 2·49-s − 4·50-s − 8·54-s + 4·62-s + 4·64-s − 6·72-s − 2·73-s + 4·75-s + 5·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4004001\)    =    \(3^{2} \cdot 23^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 4004001,\ (\ :0, 0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9568291643\)
\(L(\frac12)\)  \(\approx\)  \(0.9568291643\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;23,\;29\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - T )^{2} \)
23$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 + T^{2} \)
good2$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.226196205317528357531239238560, −9.100987847803824006949055037011, −8.890001531197521807206290837304, −8.512134568620150899138281541910, −8.133882242002105114955438928583, −7.87582107007459813947343431808, −7.31186983166383381577428667384, −7.11460933333027645543764048647, −6.95618804459985683326613097557, −5.98407845037203523055299848919, −5.93340839207471549138623551794, −4.99828753289512800788046547803, −4.65406684250847171309839188439, −3.85595751568312228144040158792, −3.57422067079487566364897976209, −3.05820833930307492662366687208, −2.63311178906281489371066347836, −2.13755445319794556580613859946, −1.44824485471940837303575561174, −0.986810074274520295692088188142, 0.986810074274520295692088188142, 1.44824485471940837303575561174, 2.13755445319794556580613859946, 2.63311178906281489371066347836, 3.05820833930307492662366687208, 3.57422067079487566364897976209, 3.85595751568312228144040158792, 4.65406684250847171309839188439, 4.99828753289512800788046547803, 5.93340839207471549138623551794, 5.98407845037203523055299848919, 6.95618804459985683326613097557, 7.11460933333027645543764048647, 7.31186983166383381577428667384, 7.87582107007459813947343431808, 8.133882242002105114955438928583, 8.512134568620150899138281541910, 8.890001531197521807206290837304, 9.100987847803824006949055037011, 9.226196205317528357531239238560

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