Properties

Label 4-21e2-1.1-c33e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $20985.5$
Root an. cond. $12.0359$
Motivic weight $33$
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  + 1.71e10·4-s − 1.58e12·7-s − 5.55e15·9-s + 2.21e20·16-s − 2.32e23·25-s − 2.73e22·28-s − 9.55e25·36-s + 1.24e26·37-s − 1.19e27·43-s − 7.72e27·49-s + 8.83e27·63-s + 2.53e30·64-s − 3.93e30·67-s − 7.64e31·79-s + 3.09e31·81-s − 3.99e33·100-s + 3.99e32·109-s − 3.51e32·112-s + 4.64e34·121-s + 127-s + 131-s + 137-s + 139-s − 1.23e36·144-s + 2.13e36·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 0.0180·7-s − 9-s + 3·16-s − 2·25-s − 0.0361·28-s − 2·36-s + 1.65·37-s − 1.33·43-s − 0.999·49-s + 0.0180·63-s + 4·64-s − 2.91·67-s − 3.73·79-s + 81-s − 4·100-s + 0.0963·109-s − 0.0542·112-s + 2·121-s − 3·144-s + 3.30·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(20985.5\)
Root analytic conductor: \(12.0359\)
Motivic weight: \(33\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(1.240683769\)
\(L(\frac12)\) \(\approx\) \(1.240683769\)
\(L(\frac{35}{2})\) 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$C_2$ \( 1 + p^{33} T^{2} \)
7$C_2$ \( 1 + 1589751452540 T + p^{33} T^{2} \)
good2$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 720032975450517890 T + p^{33} T^{2} )( 1 + 720032975450517890 T + p^{33} T^{2} ) \)
17$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
19$C_2$ \( ( 1 - \)\(15\!\cdots\!44\)\( T + p^{33} T^{2} )( 1 + \)\(15\!\cdots\!44\)\( T + p^{33} T^{2} ) \)
23$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(34\!\cdots\!08\)\( T + p^{33} T^{2} )( 1 + \)\(34\!\cdots\!08\)\( T + p^{33} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(62\!\cdots\!30\)\( T + p^{33} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
43$C_2$ \( ( 1 + \)\(59\!\cdots\!40\)\( T + p^{33} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(49\!\cdots\!18\)\( T + p^{33} T^{2} )( 1 + \)\(49\!\cdots\!18\)\( T + p^{33} T^{2} ) \)
67$C_2$ \( ( 1 + \)\(19\!\cdots\!60\)\( T + p^{33} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{33} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(10\!\cdots\!30\)\( T + p^{33} T^{2} )( 1 + \)\(10\!\cdots\!30\)\( T + p^{33} T^{2} ) \)
79$C_2$ \( ( 1 + \)\(38\!\cdots\!84\)\( T + p^{33} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{33} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(98\!\cdots\!90\)\( T + p^{33} T^{2} )( 1 + \)\(98\!\cdots\!90\)\( T + p^{33} T^{2} ) \)
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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

−11.75597968710494171526101057403, −11.36319604855647641634955305864, −11.08355023186861475243998315062, −10.03463981133916323379650735865, −10.00791387526172531542949872125, −8.882101089519224247129425864881, −8.184144687933416395144613324777, −7.64329256812074160581944276351, −7.26992270578374018833452595133, −6.39072102657627412142429426524, −6.01991648491591392154873020392, −5.70823230109624286063943868437, −4.80337174425288640686307785829, −3.85022615922826432676959765542, −3.28788307723655027109184455758, −2.64432656460013503773393470703, −2.40186040588091505622487788367, −1.47879186185359150905467252630, −1.37191212566506567442136201734, −0.18965131431482006915860453698, 0.18965131431482006915860453698, 1.37191212566506567442136201734, 1.47879186185359150905467252630, 2.40186040588091505622487788367, 2.64432656460013503773393470703, 3.28788307723655027109184455758, 3.85022615922826432676959765542, 4.80337174425288640686307785829, 5.70823230109624286063943868437, 6.01991648491591392154873020392, 6.39072102657627412142429426524, 7.26992270578374018833452595133, 7.64329256812074160581944276351, 8.184144687933416395144613324777, 8.882101089519224247129425864881, 10.00791387526172531542949872125, 10.03463981133916323379650735865, 11.08355023186861475243998315062, 11.36319604855647641634955305864, 11.75597968710494171526101057403

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