Properties

Label 4-1632e2-1.1-c1e2-0-23
Degree $4$
Conductor $2663424$
Sign $-1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 10·11-s − 2·19-s + 6·25-s − 16·41-s − 8·43-s + 8·49-s + 6·59-s + 14·67-s − 8·73-s + 81-s + 18·83-s + 8·89-s + 2·97-s − 10·99-s + 6·107-s + 10·113-s + 54·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s − 3.01·11-s − 0.458·19-s + 6/5·25-s − 2.49·41-s − 1.21·43-s + 8/7·49-s + 0.781·59-s + 1.71·67-s − 0.936·73-s + 1/9·81-s + 1.97·83-s + 0.847·89-s + 0.203·97-s − 1.00·99-s + 0.580·107-s + 0.940·113-s + 4.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2663424} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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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

−7.37988095531631371043445984952, −7.03463150345759887286555775921, −6.70233373279872832980822243158, −6.11874984930763533099921285293, −5.58652279704161692747577028199, −5.15723373714716796537666097208, −4.95987670634542136268971711709, −4.60657622347398149029938110792, −3.81856903325126906608803855517, −3.23102401602550850947175160191, −2.96095281957579972385198322248, −2.17309982560518496211728672753, −2.02330631899256921632942811500, −0.823721473581753764449057255438, 0, 0.823721473581753764449057255438, 2.02330631899256921632942811500, 2.17309982560518496211728672753, 2.96095281957579972385198322248, 3.23102401602550850947175160191, 3.81856903325126906608803855517, 4.60657622347398149029938110792, 4.95987670634542136268971711709, 5.15723373714716796537666097208, 5.58652279704161692747577028199, 6.11874984930763533099921285293, 6.70233373279872832980822243158, 7.03463150345759887286555775921, 7.37988095531631371043445984952

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