Properties

Label 4-273e2-1.1-c1e2-0-16
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·4-s + 4·7-s + 6·9-s − 12·12-s − 2·13-s + 12·16-s + 15·19-s − 12·21-s + 5·25-s − 9·27-s + 16·28-s − 15·31-s + 24·36-s − 22·37-s + 6·39-s + 8·43-s − 36·48-s + 9·49-s − 8·52-s − 45·57-s − 27·61-s + 24·63-s + 32·64-s − 11·67-s − 24·73-s − 15·75-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·4-s + 1.51·7-s + 2·9-s − 3.46·12-s − 0.554·13-s + 3·16-s + 3.44·19-s − 2.61·21-s + 25-s − 1.73·27-s + 3.02·28-s − 2.69·31-s + 4·36-s − 3.61·37-s + 0.960·39-s + 1.21·43-s − 5.19·48-s + 9/7·49-s − 1.10·52-s − 5.96·57-s − 3.45·61-s + 3.02·63-s + 4·64-s − 1.34·67-s − 2.80·73-s − 1.73·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738686864\)
\(L(\frac12)\) \(\approx\) \(1.738686864\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.87147962314324408927532650214, −11.80201489812834484710536665064, −11.05492240262172909463277311833, −11.04352919388191984109753968112, −10.39319952387523364057149836898, −10.34030112497915912392582765464, −9.358892255116999374650434056434, −8.910111944363342481758103982556, −7.75752812144092765319546872157, −7.53654773184354997369529670453, −7.11526932273722673105108455125, −6.99803199730055645006653172379, −5.91618307430411832413316018983, −5.64101106934306220767374995501, −5.18736756205102008593806676804, −4.83483730757439600057775632918, −3.56940680954281740674240294168, −2.96282131225023266566178098366, −1.55479489252007004726710845250, −1.49903082663164853734163490938, 1.49903082663164853734163490938, 1.55479489252007004726710845250, 2.96282131225023266566178098366, 3.56940680954281740674240294168, 4.83483730757439600057775632918, 5.18736756205102008593806676804, 5.64101106934306220767374995501, 5.91618307430411832413316018983, 6.99803199730055645006653172379, 7.11526932273722673105108455125, 7.53654773184354997369529670453, 7.75752812144092765319546872157, 8.910111944363342481758103982556, 9.358892255116999374650434056434, 10.34030112497915912392582765464, 10.39319952387523364057149836898, 11.04352919388191984109753968112, 11.05492240262172909463277311833, 11.80201489812834484710536665064, 11.87147962314324408927532650214

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