Properties

Label 4-21e2-1.1-c11e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $260.344$
Root an. cond. $4.01686$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 729·3-s − 2.04e3·4-s − 7.71e4·7-s + 3.54e5·9-s + 1.49e6·12-s + 2.52e7·19-s + 5.62e7·21-s + 4.88e7·25-s − 1.29e8·27-s + 1.58e8·28-s + 2.02e8·31-s − 7.25e8·36-s + 6.63e8·37-s − 3.53e9·43-s + 3.97e9·49-s − 1.83e10·57-s + 2.15e10·61-s − 2.73e10·63-s + 8.58e9·64-s + 2.14e10·67-s + 4.27e9·73-s − 3.55e10·75-s − 5.16e10·76-s − 2.14e10·79-s + 3.13e10·81-s − 1.15e11·84-s − 1.47e11·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 1.73·7-s + 2·9-s + 1.73·12-s + 2.33·19-s + 3.00·21-s + 25-s − 1.73·27-s + 1.73·28-s + 1.27·31-s − 2·36-s + 1.57·37-s − 3.66·43-s + 2.01·49-s − 4.05·57-s + 3.26·61-s − 3.47·63-s + 64-s + 1.93·67-s + 0.241·73-s − 1.73·75-s − 2.33·76-s − 0.782·79-s + 81-s − 3.00·84-s − 2.20·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(260.344\)
Root analytic conductor: \(4.01686\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.6253944317\)
\(L(\frac12)\) \(\approx\) \(0.6253944317\)
\(L(\frac{13}{2})\) 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$C_2$ \( 1 + p^{6} T + p^{11} T^{2} \)
7$C_2$ \( 1 + 77153 T + p^{11} T^{2} \)
good2$C_2^2$ \( 1 + p^{11} T^{2} + p^{22} T^{4} \)
5$C_2^2$ \( 1 - p^{11} T^{2} + p^{22} T^{4} \)
11$C_2^2$ \( 1 + p^{11} T^{2} + p^{22} T^{4} \)
13$C_2$ \( ( 1 - 2382989 T + p^{11} T^{2} )( 1 + 2382989 T + p^{11} T^{2} ) \)
17$C_2^2$ \( 1 - p^{11} T^{2} + p^{22} T^{4} \)
19$C_2$ \( ( 1 - 20581901 T + p^{11} T^{2} )( 1 - 4655368 T + p^{11} T^{2} ) \)
23$C_2^2$ \( 1 + p^{11} T^{2} + p^{22} T^{4} \)
29$C_2$ \( ( 1 - p^{11} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 249734764 T + p^{11} T^{2} )( 1 + 46742179 T + p^{11} T^{2} ) \)
37$C_2$ \( ( 1 - 782919730 T + p^{11} T^{2} )( 1 + 119374607 T + p^{11} T^{2} ) \)
41$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 1768358135 T + p^{11} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{11} T^{2} + p^{22} T^{4} \)
53$C_2^2$ \( 1 + p^{11} T^{2} + p^{22} T^{4} \)
59$C_2^2$ \( 1 - p^{11} T^{2} + p^{22} T^{4} \)
61$C_2$ \( ( 1 - 12977292913 T + p^{11} T^{2} )( 1 - 8546352539 T + p^{11} T^{2} ) \)
67$C_2$ \( ( 1 - 15458751248 T + p^{11} T^{2} )( 1 - 5951291615 T + p^{11} T^{2} ) \)
71$C_2$ \( ( 1 - p^{11} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 19805520230 T + p^{11} T^{2} )( 1 + 15533161001 T + p^{11} T^{2} ) \)
79$C_2$ \( ( 1 - 32885832404 T + p^{11} T^{2} )( 1 + 54296224537 T + p^{11} T^{2} ) \)
83$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{11} T^{2} + p^{22} T^{4} \)
97$C_2$ \( ( 1 - 112637211442 T + p^{11} T^{2} )( 1 + 112637211442 T + p^{11} T^{2} ) \)
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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

−15.87120685711821636619520165145, −15.70239018114830833869061258745, −14.48745541171504812652862850540, −13.54625619543939111656750465412, −13.14209106355966646318866666612, −12.66519505913501434828279708948, −11.68185929555701624873536147365, −11.52488076636173629532336542752, −10.17615987994714299937514811430, −9.886125155080246848289007398095, −9.354901834655399639726450017452, −8.183724634892179577670966637019, −6.76196309098378319371975946692, −6.69072964154620511027888855633, −5.44999995668248908107810621960, −5.05403879296783702763291689479, −3.95667666988824479989651290073, −2.99152536230134208411209198025, −0.984079518132159641243900442276, −0.47020608590198100993697020433, 0.47020608590198100993697020433, 0.984079518132159641243900442276, 2.99152536230134208411209198025, 3.95667666988824479989651290073, 5.05403879296783702763291689479, 5.44999995668248908107810621960, 6.69072964154620511027888855633, 6.76196309098378319371975946692, 8.183724634892179577670966637019, 9.354901834655399639726450017452, 9.886125155080246848289007398095, 10.17615987994714299937514811430, 11.52488076636173629532336542752, 11.68185929555701624873536147365, 12.66519505913501434828279708948, 13.14209106355966646318866666612, 13.54625619543939111656750465412, 14.48745541171504812652862850540, 15.70239018114830833869061258745, 15.87120685711821636619520165145

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