Properties

Label 4-21e3-1.1-c1e2-0-1
Degree $4$
Conductor $9261$
Sign $-1$
Analytic cond. $0.590489$
Root an. cond. $0.876603$
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 − 4·5-s − 7-s + 9-s − 3·12-s − 4·15-s + 5·16-s − 12·17-s + 12·20-s − 21-s + 2·25-s + 27-s + 3·28-s + 4·35-s − 3·36-s + 12·37-s + 4·41-s − 8·43-s − 4·45-s + 5·48-s + 49-s − 12·51-s + 24·59-s + 12·60-s − 63-s − 3·64-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.866·12-s − 1.03·15-s + 5/4·16-s − 2.91·17-s + 2.68·20-s − 0.218·21-s + 2/5·25-s + 0.192·27-s + 0.566·28-s + 0.676·35-s − 1/2·36-s + 1.97·37-s + 0.624·41-s − 1.21·43-s − 0.596·45-s + 0.721·48-s + 1/7·49-s − 1.68·51-s + 3.12·59-s + 1.54·60-s − 0.125·63-s − 3/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9261\)    =    \(3^{3} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(0.590489\)
Root analytic conductor: \(0.876603\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9261} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 9261,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
7$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−11.55595081754559805664353277943, −10.92437047522813718231601421137, −9.800766905340478341584155035885, −9.724661210726173973340366981146, −8.672365196683144779961098572501, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −7.25047783802838427330028450541, −6.81422856027217448872168795499, −5.73267711633132091527857634664, −4.61914309368534933978095799917, −4.13559084050773741974089362056, −3.94328917656394473634657471263, −2.64839746138688614832708523229, 0, 2.64839746138688614832708523229, 3.94328917656394473634657471263, 4.13559084050773741974089362056, 4.61914309368534933978095799917, 5.73267711633132091527857634664, 6.81422856027217448872168795499, 7.25047783802838427330028450541, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 8.672365196683144779961098572501, 9.724661210726173973340366981146, 9.800766905340478341584155035885, 10.92437047522813718231601421137, 11.55595081754559805664353277943

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