Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 4·7-s + 3·11-s + 13-s + 3·17-s − 7·19-s + 9·23-s + 5·25-s + 3·29-s − 16·31-s + 12·35-s + 37-s + 3·41-s − 43-s + 9·49-s + 3·53-s + 9·55-s + 4·61-s + 3·65-s + 8·67-s + 24·71-s − 11·73-s + 12·77-s + 32·79-s + 9·83-s + 9·85-s + 3·89-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.51·7-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 1.60·19-s + 1.87·23-s + 25-s + 0.557·29-s − 2.87·31-s + 2.02·35-s + 0.164·37-s + 0.468·41-s − 0.152·43-s + 9/7·49-s + 0.412·53-s + 1.21·55-s + 0.512·61-s + 0.372·65-s + 0.977·67-s + 2.84·71-s − 1.28·73-s + 1.36·77-s + 3.60·79-s + 0.987·83-s + 0.976·85-s + 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.656787566\)
\(L(\frac12)\) \(\approx\) \(5.656787566\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−8.838851042366998806900759130497, −8.822878100594114865945204088360, −8.059120169174377305584619459297, −7.964870370083415265103848481142, −7.35597706933006908832816689482, −7.02064927995795923452348912758, −6.54054017613508464442419471293, −6.35021738247783955624437408215, −5.85705816305107194735936284920, −5.25443049034497508827519114832, −5.18065281311061335587419124054, −4.89398231997440069608512627800, −4.19204225946770728682520687994, −3.75368662891185918176227189437, −3.47690111054046174340751629999, −2.62298183179987522283068605653, −2.11695863038468567091529565257, −1.90845984032894267684025546185, −1.26717037014811580907718359026, −0.815088068354780966576537024946, 0.815088068354780966576537024946, 1.26717037014811580907718359026, 1.90845984032894267684025546185, 2.11695863038468567091529565257, 2.62298183179987522283068605653, 3.47690111054046174340751629999, 3.75368662891185918176227189437, 4.19204225946770728682520687994, 4.89398231997440069608512627800, 5.18065281311061335587419124054, 5.25443049034497508827519114832, 5.85705816305107194735936284920, 6.35021738247783955624437408215, 6.54054017613508464442419471293, 7.02064927995795923452348912758, 7.35597706933006908832816689482, 7.964870370083415265103848481142, 8.059120169174377305584619459297, 8.822878100594114865945204088360, 8.838851042366998806900759130497

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