Properties

Label 4-21e2-1.1-c39e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $40930.5$
Root an. cond. $14.2236$
Motivic weight $39$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.09e12·4-s − 5.45e16·7-s − 4.05e18·9-s + 9.06e23·16-s − 3.63e27·25-s − 6.00e28·28-s − 4.45e30·36-s + 1.03e30·37-s + 1.24e32·43-s + 2.07e33·49-s + 2.21e35·63-s + 6.64e35·64-s + 7.22e35·67-s + 2.18e37·79-s + 1.64e37·81-s − 3.99e39·100-s − 1.21e40·109-s − 4.95e40·112-s + 8.22e40·121-s + 127-s + 131-s + 137-s + 139-s − 3.67e42·144-s + 1.13e42·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 1.81·7-s − 9-s + 3·16-s − 2·25-s − 3.62·28-s − 2·36-s + 0.271·37-s + 1.75·43-s + 2.27·49-s + 1.81·63-s + 4·64-s + 1.78·67-s + 2.16·79-s + 81-s − 4·100-s − 2.25·109-s − 5.43·112-s + 2·121-s − 3·144-s + 0.542·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+39/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(40930.5\)
Root analytic conductor: \(14.2236\)
Motivic weight: \(39\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :39/2, 39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(4.964579402\)
\(L(\frac12)\) \(\approx\) \(4.964579402\)
\(L(\frac{41}{2})\) 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$C_2$ \( 1 + p^{39} T^{2} \)
7$C_2$ \( 1 + 54595696320612736 T + p^{39} T^{2} \)
good2$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
13$C_2$ \( ( 1 - \)\(10\!\cdots\!10\)\( T + p^{39} T^{2} )( 1 + \)\(10\!\cdots\!10\)\( T + p^{39} T^{2} ) \)
17$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
19$C_2$ \( ( 1 - \)\(16\!\cdots\!04\)\( T + p^{39} T^{2} )( 1 + \)\(16\!\cdots\!04\)\( T + p^{39} T^{2} ) \)
23$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(23\!\cdots\!28\)\( T + p^{39} T^{2} )( 1 + \)\(23\!\cdots\!28\)\( T + p^{39} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(51\!\cdots\!30\)\( T + p^{39} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
43$C_2$ \( ( 1 - \)\(62\!\cdots\!40\)\( T + p^{39} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(78\!\cdots\!42\)\( T + p^{39} T^{2} )( 1 + \)\(78\!\cdots\!42\)\( T + p^{39} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(36\!\cdots\!40\)\( T + p^{39} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(29\!\cdots\!70\)\( T + p^{39} T^{2} )( 1 + \)\(29\!\cdots\!70\)\( T + p^{39} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(10\!\cdots\!24\)\( T + p^{39} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(60\!\cdots\!90\)\( T + p^{39} T^{2} )( 1 + \)\(60\!\cdots\!90\)\( T + p^{39} T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08032216004088183742248738201, −11.06141554185866657944236902544, −10.18941107291136916475295904162, −9.724489876390417096211203571782, −9.248608114800294806487062099725, −8.293243868561784737823190532676, −7.75415805842596426854711322157, −7.24996454380325352572065411446, −6.54515711977350486475237043970, −6.35020690633978264816665195188, −5.61255075580973117695747334206, −5.60082413014624575458279490187, −4.15090282613603660968882864534, −3.51476135388203516136593097706, −3.21990280785127080963323779897, −2.55958447963267854874124540278, −2.26873826374979445380712122208, −1.72030802078574462773256295154, −0.68474147091730658447151654117, −0.52858109233238948700668611307, 0.52858109233238948700668611307, 0.68474147091730658447151654117, 1.72030802078574462773256295154, 2.26873826374979445380712122208, 2.55958447963267854874124540278, 3.21990280785127080963323779897, 3.51476135388203516136593097706, 4.15090282613603660968882864534, 5.60082413014624575458279490187, 5.61255075580973117695747334206, 6.35020690633978264816665195188, 6.54515711977350486475237043970, 7.24996454380325352572065411446, 7.75415805842596426854711322157, 8.293243868561784737823190532676, 9.248608114800294806487062099725, 9.724489876390417096211203571782, 10.18941107291136916475295904162, 11.06141554185866657944236902544, 11.08032216004088183742248738201

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