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Properties

Label 4-21e2-1.1-c38e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $36892.1$
Root an. cond. $13.8590$
Motivic weight $38$
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  + 1.16e9·3-s − 2.74e11·4-s − 1.34e16·7-s − 3.19e20·12-s − 5.72e21·13-s − 2.55e24·19-s − 1.55e25·21-s − 3.63e26·25-s − 1.57e27·27-s + 3.68e27·28-s + 3.31e28·31-s − 1.24e30·37-s − 6.65e30·39-s + 2.88e31·43-s + 5.00e31·49-s + 1.57e33·52-s − 2.97e33·57-s − 3.04e33·61-s + 2.07e34·64-s − 9.54e34·67-s − 4.78e35·73-s − 4.22e35·75-s + 7.03e35·76-s + 2.20e36·79-s − 1.82e36·81-s + 4.28e36·84-s + 7.67e37·91-s + ⋯
L(s)  = 1  + 3-s − 4-s − 1.17·7-s − 12-s − 3.91·13-s − 1.29·19-s − 1.17·21-s − 25-s − 27-s + 1.17·28-s + 1.53·31-s − 1.99·37-s − 3.91·39-s + 2.65·43-s + 0.385·49-s + 3.91·52-s − 1.29·57-s − 0.365·61-s + 64-s − 1.92·67-s − 1.89·73-s − 75-s + 1.29·76-s + 1.93·79-s − 81-s + 1.17·84-s + 4.60·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+19)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36892.1\)
Root analytic conductor: \(13.8590\)
Motivic weight: \(38\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :19, 19),\ 1)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.6028897306\)
\(L(\frac12)\) \(\approx\) \(0.6028897306\)
\(L(20)\) 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$C_2$ \( 1 - p^{19} T + p^{38} T^{2} \)
7$C_2$ \( 1 + 13416617883005076 T + p^{38} T^{2} \)
good2$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
5$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
11$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
13$C_2$ \( ( 1 + \)\(28\!\cdots\!13\)\( T + p^{38} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
19$C_2$ \( ( 1 - \)\(13\!\cdots\!83\)\( T + p^{38} T^{2} )( 1 + \)\(38\!\cdots\!74\)\( T + p^{38} T^{2} ) \)
23$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{19} T )^{2}( 1 + p^{19} T )^{2} \)
31$C_2$ \( ( 1 - \)\(40\!\cdots\!39\)\( T + p^{38} T^{2} )( 1 + \)\(75\!\cdots\!06\)\( T + p^{38} T^{2} ) \)
37$C_2$ \( ( 1 + \)\(60\!\cdots\!17\)\( T + p^{38} T^{2} )( 1 + \)\(64\!\cdots\!46\)\( T + p^{38} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{19} T )^{2}( 1 + p^{19} T )^{2} \)
43$C_2$ \( ( 1 - \)\(14\!\cdots\!11\)\( T + p^{38} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
53$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
59$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
61$C_2$ \( ( 1 - \)\(12\!\cdots\!19\)\( T + p^{38} T^{2} )( 1 + \)\(15\!\cdots\!93\)\( T + p^{38} T^{2} ) \)
67$C_2$ \( ( 1 + \)\(24\!\cdots\!42\)\( T + p^{38} T^{2} )( 1 + \)\(71\!\cdots\!81\)\( T + p^{38} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{19} T )^{2}( 1 + p^{19} T )^{2} \)
73$C_2$ \( ( 1 + \)\(97\!\cdots\!74\)\( T + p^{38} T^{2} )( 1 + \)\(38\!\cdots\!93\)\( T + p^{38} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(15\!\cdots\!38\)\( T + p^{38} T^{2} )( 1 - \)\(61\!\cdots\!71\)\( T + p^{38} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{19} T )^{2}( 1 + p^{19} T )^{2} \)
89$C_2$ \( ( 1 - p^{19} T + p^{38} T^{2} )( 1 + p^{19} T + p^{38} T^{2} ) \)
97$C_2$ \( ( 1 + \)\(21\!\cdots\!62\)\( T + p^{38} T^{2} )^{2} \)
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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

−11.68656282945293843985838435032, −10.29957295485537133119845704223, −10.21163016893098419215768339786, −9.566409412547901375197320531422, −9.077251601684503824342081143864, −8.902282816556849691057469329704, −7.76346563571109002474958023071, −7.68614648887974239801070963346, −6.92778550447498053515056039871, −6.32834935927607390839177547962, −5.45857379868393793340062670222, −4.95636897620041493404662178539, −4.29852423094626223151791879661, −4.06043809879431966935653582558, −3.04712713904990880701969431492, −2.70588193408403579238082677011, −2.32117569178537157199230454261, −1.78361661522094495934479205683, −0.42937640956005989039006397452, −0.27660912583676661030245985425, 0.27660912583676661030245985425, 0.42937640956005989039006397452, 1.78361661522094495934479205683, 2.32117569178537157199230454261, 2.70588193408403579238082677011, 3.04712713904990880701969431492, 4.06043809879431966935653582558, 4.29852423094626223151791879661, 4.95636897620041493404662178539, 5.45857379868393793340062670222, 6.32834935927607390839177547962, 6.92778550447498053515056039871, 7.68614648887974239801070963346, 7.76346563571109002474958023071, 8.902282816556849691057469329704, 9.077251601684503824342081143864, 9.566409412547901375197320531422, 10.21163016893098419215768339786, 10.29957295485537133119845704223, 11.68656282945293843985838435032

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