Properties

Label 4-21e2-1.1-c20e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $2834.27$
Root an. cond. $7.29642$
Motivic weight $20$
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  − 5.90e4·3-s − 1.04e6·4-s − 4.45e8·7-s + 6.19e10·12-s + 7.63e10·13-s − 1.20e13·19-s + 2.63e13·21-s − 9.53e13·25-s + 2.05e14·27-s + 4.67e14·28-s − 8.45e14·31-s + 7.21e14·37-s − 4.50e15·39-s + 2.68e15·43-s + 1.19e17·49-s − 8.00e16·52-s + 7.08e17·57-s + 1.38e18·61-s + 1.15e18·64-s + 2.05e18·67-s + 4.76e18·73-s + 5.63e18·75-s + 1.25e19·76-s + 1.45e19·79-s − 1.21e19·81-s − 2.76e19·84-s − 3.40e19·91-s + ⋯
L(s)  = 1  − 3-s − 4-s − 1.57·7-s + 12-s + 0.553·13-s − 1.95·19-s + 1.57·21-s − 25-s + 27-s + 1.57·28-s − 1.03·31-s + 0.149·37-s − 0.553·39-s + 0.124·43-s + 1.49·49-s − 0.553·52-s + 1.95·57-s + 1.94·61-s + 64-s + 1.12·67-s + 1.10·73-s + 75-s + 1.95·76-s + 1.53·79-s − 81-s − 1.57·84-s − 0.874·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+10)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2834.27\)
Root analytic conductor: \(7.29642\)
Motivic weight: \(20\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :10, 10),\ 1)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.3311588825\)
\(L(\frac12)\) \(\approx\) \(0.3311588825\)
\(L(11)\) 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$C_2$ \( 1 + p^{10} T + p^{20} T^{2} \)
7$C_2$ \( 1 + 445987849 T + p^{20} T^{2} \)
good2$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
5$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
11$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
13$C_2$ \( ( 1 - 38170922351 T + p^{20} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
19$C_2$ \( ( 1 + 3843000838126 T + p^{20} T^{2} )( 1 + 8162815561273 T + p^{20} T^{2} ) \)
23$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
31$C_2$ \( ( 1 - 793864193940674 T + p^{20} T^{2} )( 1 + 1638989389385401 T + p^{20} T^{2} ) \)
37$C_2$ \( ( 1 - 8665815522315698 T + p^{20} T^{2} )( 1 + 7944719218014049 T + p^{20} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
43$C_2$ \( ( 1 - 1343935055601623 T + p^{20} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
53$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
59$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
61$C_2$ \( ( 1 - 980437652641223327 T + p^{20} T^{2} )( 1 - 407350578138766799 T + p^{20} T^{2} ) \)
67$C_2$ \( ( 1 - 3635284414544796023 T + p^{20} T^{2} )( 1 + 1579433466595168174 T + p^{20} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
73$C_2$ \( ( 1 - 8577821547816235298 T + p^{20} T^{2} )( 1 + 3815100224458033249 T + p^{20} T^{2} ) \)
79$C_2$ \( ( 1 - 17785677669406834199 T + p^{20} T^{2} )( 1 + 3262829215661625598 T + p^{20} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
89$C_2$ \( ( 1 - p^{10} T + p^{20} T^{2} )( 1 + p^{10} T + p^{20} T^{2} ) \)
97$C_2$ \( ( 1 + \)\(14\!\cdots\!74\)\( T + p^{20} T^{2} )^{2} \)
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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

−13.48745082352883059669965236999, −13.37190232073269719052296312295, −12.49393364062664988364948648482, −12.29420958066693377431871922042, −11.06359778754607027220787069768, −10.87803783619346733736939368996, −9.849818151163529075161710952050, −9.489054283966129902935339699730, −8.682544840052064427895371799639, −8.154622021838432889349669442474, −6.74842512205168935321612887885, −6.57204580922313652210053713294, −5.71532758575011323510913237097, −5.23315432170616270812467327422, −4.03888092472912438136834025894, −3.94025732225577878571247253849, −2.80936034111701620158083430381, −1.94917754515467158211327362728, −0.69712901078600067047144790160, −0.25564856661877053300332969247, 0.25564856661877053300332969247, 0.69712901078600067047144790160, 1.94917754515467158211327362728, 2.80936034111701620158083430381, 3.94025732225577878571247253849, 4.03888092472912438136834025894, 5.23315432170616270812467327422, 5.71532758575011323510913237097, 6.57204580922313652210053713294, 6.74842512205168935321612887885, 8.154622021838432889349669442474, 8.682544840052064427895371799639, 9.489054283966129902935339699730, 9.849818151163529075161710952050, 10.87803783619346733736939368996, 11.06359778754607027220787069768, 12.29420958066693377431871922042, 12.49393364062664988364948648482, 13.37190232073269719052296312295, 13.48745082352883059669965236999

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