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Properties

Label 4-21e2-1.1-c25e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $6915.47$
Root an. cond. $9.11917$
Motivic weight $25$
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  − 1.59e6·3-s − 3.35e7·4-s − 7.31e10·7-s + 1.69e12·9-s + 5.34e13·12-s + 8.46e15·19-s + 1.16e17·21-s + 2.98e17·25-s − 1.35e18·27-s + 2.45e18·28-s − 4.26e18·31-s − 5.68e19·36-s − 7.07e19·37-s + 3.63e20·43-s + 4.01e21·49-s − 1.34e22·57-s − 6.32e22·61-s − 1.24e23·63-s + 3.77e22·64-s + 5.57e22·67-s + 2.91e23·73-s − 4.75e23·75-s − 2.83e23·76-s + 1.04e24·79-s + 7.17e23·81-s − 3.91e24·84-s + 6.79e24·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 1.99·7-s + 2·9-s + 1.73·12-s + 0.877·19-s + 3.46·21-s + 25-s − 1.73·27-s + 1.99·28-s − 0.971·31-s − 2·36-s − 1.76·37-s + 1.38·43-s + 2.99·49-s − 1.51·57-s − 3.05·61-s − 3.99·63-s + 64-s + 0.832·67-s + 1.48·73-s − 1.73·75-s − 0.877·76-s + 1.99·79-s + 81-s − 3.46·84-s + 1.68·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6915.47\)
Root analytic conductor: \(9.11917\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(0.2057278671\)
\(L(\frac12)\) \(\approx\) \(0.2057278671\)
\(L(\frac{27}{2})\) 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$C_2$ \( 1 + p^{13} T + p^{25} T^{2} \)
7$C_2$ \( 1 + 73180401839 T + p^{25} T^{2} \)
good2$C_2^2$ \( 1 + p^{25} T^{2} + p^{50} T^{4} \)
5$C_2^2$ \( 1 - p^{25} T^{2} + p^{50} T^{4} \)
11$C_2^2$ \( 1 + p^{25} T^{2} + p^{50} T^{4} \)
13$C_2$ \( ( 1 - 163988887345927 T + p^{25} T^{2} )( 1 + 163988887345927 T + p^{25} T^{2} ) \)
17$C_2^2$ \( 1 - p^{25} T^{2} + p^{50} T^{4} \)
19$C_2$ \( ( 1 - 13564834656085432 T + p^{25} T^{2} )( 1 + 5101434430943761 T + p^{25} T^{2} ) \)
23$C_2^2$ \( 1 + p^{25} T^{2} + p^{50} T^{4} \)
29$C_2$ \( ( 1 - p^{25} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2078341811654865124 T + p^{25} T^{2} )( 1 + 6340446222129779749 T + p^{25} T^{2} ) \)
37$C_2$ \( ( 1 + 2969395772201258411 T + p^{25} T^{2} )( 1 + 67823314319585129750 T + p^{25} T^{2} ) \)
41$C_2$ \( ( 1 + p^{25} T^{2} )^{2} \)
43$C_2$ \( ( 1 - \)\(18\!\cdots\!75\)\( T + p^{25} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{25} T^{2} + p^{50} T^{4} \)
53$C_2^2$ \( 1 + p^{25} T^{2} + p^{50} T^{4} \)
59$C_2^2$ \( 1 - p^{25} T^{2} + p^{50} T^{4} \)
61$C_2$ \( ( 1 + \)\(21\!\cdots\!27\)\( T + p^{25} T^{2} )( 1 + \)\(41\!\cdots\!01\)\( T + p^{25} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(13\!\cdots\!25\)\( T + p^{25} T^{2} )( 1 + \)\(77\!\cdots\!36\)\( T + p^{25} T^{2} ) \)
71$C_2$ \( ( 1 - p^{25} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(32\!\cdots\!23\)\( T + p^{25} T^{2} )( 1 + \)\(31\!\cdots\!50\)\( T + p^{25} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(60\!\cdots\!36\)\( T + p^{25} T^{2} )( 1 - \)\(44\!\cdots\!57\)\( T + p^{25} T^{2} ) \)
83$C_2$ \( ( 1 + p^{25} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{25} T^{2} + p^{50} T^{4} \)
97$C_2$ \( ( 1 - \)\(10\!\cdots\!14\)\( T + p^{25} T^{2} )( 1 + \)\(10\!\cdots\!14\)\( T + p^{25} T^{2} ) \)
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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

−12.80820630644158990062138361945, −12.32958102987916817447629529291, −12.14520814469825071149309938846, −10.93769588542310267166742520558, −10.66877993286668557148643295734, −9.909922353528808902216740778752, −9.254067331809483018942902086427, −9.089860177433482285979751594375, −7.76415886095022502637581318901, −6.82631240265340800825864339723, −6.70054823345590201061238253468, −5.81817265586083186642796244257, −5.35335217461519355678739971063, −4.76258548085961490439840660594, −3.93243102676817272886013399893, −3.44887611969264203812925026876, −2.57738821313122023299998347074, −1.38301641550212403699801117561, −0.70259947698886766915235226779, −0.19330195581018203165882063752, 0.19330195581018203165882063752, 0.70259947698886766915235226779, 1.38301641550212403699801117561, 2.57738821313122023299998347074, 3.44887611969264203812925026876, 3.93243102676817272886013399893, 4.76258548085961490439840660594, 5.35335217461519355678739971063, 5.81817265586083186642796244257, 6.70054823345590201061238253468, 6.82631240265340800825864339723, 7.76415886095022502637581318901, 9.089860177433482285979751594375, 9.254067331809483018942902086427, 9.909922353528808902216740778752, 10.66877993286668557148643295734, 10.93769588542310267166742520558, 12.14520814469825071149309938846, 12.32958102987916817447629529291, 12.80820630644158990062138361945

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