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Properties

Label 4-4563-1.1-c1e2-0-1
Degree $4$
Conductor $4563$
Sign $-1$
Analytic cond. $0.290940$
Root an. cond. $0.734431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s − 8·7-s + 9-s + 3·12-s + 2·13-s + 5·16-s + 8·21-s − 6·25-s − 27-s + 24·28-s + 8·31-s − 3·36-s − 4·37-s − 2·39-s − 24·43-s − 5·48-s + 34·49-s − 6·52-s − 4·61-s − 8·63-s − 3·64-s − 16·67-s + 4·73-s + 6·75-s + 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s − 3.02·7-s + 1/3·9-s + 0.866·12-s + 0.554·13-s + 5/4·16-s + 1.74·21-s − 6/5·25-s − 0.192·27-s + 4.53·28-s + 1.43·31-s − 1/2·36-s − 0.657·37-s − 0.320·39-s − 3.65·43-s − 0.721·48-s + 34/7·49-s − 0.832·52-s − 0.512·61-s − 1.00·63-s − 3/8·64-s − 1.95·67-s + 0.468·73-s + 0.692·75-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(0.290940\)
Root analytic conductor: \(0.734431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4563,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 + T \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−12.03518823883711509314697782796, −11.85387650647454943465550811155, −10.56874363227147435358239986991, −9.952802346640542280811898231944, −9.880880528421929637795747545010, −9.192740554674130435254875131876, −8.676495022208841081429592670025, −7.80846844879300530070948409490, −6.54556545042783689480045128663, −6.52300807259623823318085089821, −5.65486780279705070512904066158, −4.73928185911620471935520190469, −3.71205447024015551816570926916, −3.25802804921991097162798742437, 0, 3.25802804921991097162798742437, 3.71205447024015551816570926916, 4.73928185911620471935520190469, 5.65486780279705070512904066158, 6.52300807259623823318085089821, 6.54556545042783689480045128663, 7.80846844879300530070948409490, 8.676495022208841081429592670025, 9.192740554674130435254875131876, 9.880880528421929637795747545010, 9.952802346640542280811898231944, 10.56874363227147435358239986991, 11.85387650647454943465550811155, 12.03518823883711509314697782796

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