Properties

Degree 4
Conductor $ 3^{2} \cdot 11^{2} \cdot 23 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·4-s − 5-s + 6·9-s + 3·11-s + 9·12-s + 3·15-s + 5·16-s + 3·20-s + 7·23-s − 7·25-s − 9·27-s − 5·31-s − 9·33-s − 18·36-s − 3·37-s − 9·44-s − 6·45-s + 5·47-s − 15·48-s − 11·49-s − 5·53-s − 3·55-s − 3·59-s − 9·60-s − 3·64-s − 6·67-s + ⋯
L(s)  = 1  − 1.73·3-s − 3/2·4-s − 0.447·5-s + 2·9-s + 0.904·11-s + 2.59·12-s + 0.774·15-s + 5/4·16-s + 0.670·20-s + 1.45·23-s − 7/5·25-s − 1.73·27-s − 0.898·31-s − 1.56·33-s − 3·36-s − 0.493·37-s − 1.35·44-s − 0.894·45-s + 0.729·47-s − 2.16·48-s − 1.57·49-s − 0.686·53-s − 0.404·55-s − 0.390·59-s − 1.16·60-s − 3/8·64-s − 0.733·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(25047\)    =    \(3^{2} \cdot 11^{2} \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{25047} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 25047,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 143 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−10.45715402421865550017158147261, −9.907813772532039625194893482897, −9.354255600192595666072923996922, −8.987334907847032372266694963918, −8.355452370493371975346300152634, −7.47545237040884299028314839664, −7.16275471097725371864787753723, −6.22985218533883797555173099994, −5.89850219230318835555125348726, −5.06005851722607127975323573237, −4.73102153871470454016218385864, −4.08241272011850993008377291555, −3.44170186527149644991256333227, −1.38142750019945230118042835170, 0, 1.38142750019945230118042835170, 3.44170186527149644991256333227, 4.08241272011850993008377291555, 4.73102153871470454016218385864, 5.06005851722607127975323573237, 5.89850219230318835555125348726, 6.22985218533883797555173099994, 7.16275471097725371864787753723, 7.47545237040884299028314839664, 8.355452370493371975346300152634, 8.987334907847032372266694963918, 9.354255600192595666072923996922, 9.907813772532039625194893482897, 10.45715402421865550017158147261

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