Properties

Label 4-21e2-1.1-c40e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $45291.8$
Root an. cond. $14.5883$
Motivic weight $40$
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  − 3.48e9·3-s − 1.09e12·4-s + 3.93e16·7-s + 3.83e21·12-s − 7.31e22·13-s − 6.89e25·19-s − 1.37e26·21-s − 9.09e27·25-s + 4.23e28·27-s − 4.32e28·28-s + 6.29e29·31-s + 4.57e31·37-s + 2.54e32·39-s − 1.86e33·43-s − 4.82e33·49-s + 8.03e34·52-s + 2.40e35·57-s − 9.08e35·61-s + 1.32e36·64-s + 2.41e36·67-s + 1.42e37·73-s + 3.17e37·75-s + 7.58e37·76-s − 3.16e37·79-s − 1.47e38·81-s + 1.50e38·84-s − 2.87e39·91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 0.492·7-s + 12-s − 3.84·13-s − 1.83·19-s − 0.492·21-s − 25-s + 27-s − 0.492·28-s + 0.936·31-s + 1.97·37-s + 3.84·39-s − 3.99·43-s − 0.757·49-s + 3.84·52-s + 1.83·57-s − 1.78·61-s + 64-s + 0.727·67-s + 0.771·73-s + 75-s + 1.83·76-s − 0.352·79-s − 81-s + 0.492·84-s − 1.89·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+20)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(45291.8\)
Root analytic conductor: \(14.5883\)
Motivic weight: \(40\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :20, 20),\ 1)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(0.2567390961\)
\(L(\frac12)\) \(\approx\) \(0.2567390961\)
\(L(21)\) 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$C_2$ \( 1 + p^{20} T + p^{40} T^{2} \)
7$C_2$ \( 1 - 39320628860422800 T + p^{40} T^{2} \)
good2$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
5$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
11$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
13$C_2$ \( ( 1 + \)\(36\!\cdots\!01\)\( T + p^{40} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
19$C_2$ \( ( 1 + \)\(85\!\cdots\!73\)\( T + p^{40} T^{2} )( 1 + \)\(60\!\cdots\!26\)\( T + p^{40} T^{2} ) \)
23$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{20} T )^{2}( 1 + p^{20} T )^{2} \)
31$C_2$ \( ( 1 - \)\(13\!\cdots\!99\)\( T + p^{40} T^{2} )( 1 + \)\(71\!\cdots\!26\)\( T + p^{40} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(28\!\cdots\!02\)\( T + p^{40} T^{2} )( 1 - \)\(16\!\cdots\!99\)\( T + p^{40} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{20} T )^{2}( 1 + p^{20} T )^{2} \)
43$C_2$ \( ( 1 + \)\(93\!\cdots\!73\)\( T + p^{40} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
53$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
59$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
61$C_2$ \( ( 1 + \)\(56\!\cdots\!73\)\( T + p^{40} T^{2} )( 1 + \)\(85\!\cdots\!01\)\( T + p^{40} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(65\!\cdots\!27\)\( T + p^{40} T^{2} )( 1 + \)\(41\!\cdots\!26\)\( T + p^{40} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{20} T )^{2}( 1 + p^{20} T )^{2} \)
73$C_2$ \( ( 1 - \)\(36\!\cdots\!02\)\( T + p^{40} T^{2} )( 1 + \)\(22\!\cdots\!01\)\( T + p^{40} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(13\!\cdots\!99\)\( T + p^{40} T^{2} )( 1 + \)\(16\!\cdots\!98\)\( T + p^{40} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{20} T )^{2}( 1 + p^{20} T )^{2} \)
89$C_2$ \( ( 1 - p^{20} T + p^{40} T^{2} )( 1 + p^{20} T + p^{40} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(10\!\cdots\!74\)\( T + p^{40} T^{2} )^{2} \)
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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

−11.41228011742615635419879500045, −10.56576771991268565740923633870, −9.925209436365992630762187565304, −9.795930727703428107175642101971, −9.100628285859780441724535281976, −8.247166687802911032388498144196, −7.952013836118456364500283061504, −7.24310732322409152547364960980, −6.55182184330857643523435020924, −6.18404207871171335409812123101, −5.10316967526853517898843160731, −4.98752868265940202214269557792, −4.66558057168868896953643085496, −4.19047607726869185562613734658, −3.18725160151216245245618082681, −2.32591173483457708905891533144, −2.28802849856753639709373396592, −1.40682969018832242539461663979, −0.35094605161172732436381225610, −0.25862881723275194764238812164, 0.25862881723275194764238812164, 0.35094605161172732436381225610, 1.40682969018832242539461663979, 2.28802849856753639709373396592, 2.32591173483457708905891533144, 3.18725160151216245245618082681, 4.19047607726869185562613734658, 4.66558057168868896953643085496, 4.98752868265940202214269557792, 5.10316967526853517898843160731, 6.18404207871171335409812123101, 6.55182184330857643523435020924, 7.24310732322409152547364960980, 7.952013836118456364500283061504, 8.247166687802911032388498144196, 9.100628285859780441724535281976, 9.795930727703428107175642101971, 9.925209436365992630762187565304, 10.56576771991268565740923633870, 11.41228011742615635419879500045

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