Properties

Label 4-21e2-1.1-c3e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $1.53522$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·4-s − 20·7-s − 27·9-s + 192·16-s − 250·25-s − 320·28-s − 432·36-s − 220·37-s + 1.04e3·43-s + 57·49-s + 540·63-s + 2.04e3·64-s − 1.76e3·67-s + 1.76e3·79-s + 729·81-s − 4.00e3·100-s + 1.29e3·109-s − 3.84e3·112-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s − 5.18e3·144-s − 3.52e3·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 1.07·7-s − 9-s + 3·16-s − 2·25-s − 2.15·28-s − 2·36-s − 0.977·37-s + 3.68·43-s + 0.166·49-s + 1.07·63-s + 4·64-s − 3.20·67-s + 2.51·79-s + 81-s − 4·100-s + 1.13·109-s − 3.23·112-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3·144-s − 1.95·148-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.53522\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.458362107\)
\(L(\frac12)\) \(\approx\) \(1.458362107\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{3} T^{2} \)
7$C_2$ \( 1 + 20 T + p^{3} T^{2} \)
good2$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 56 T + p^{3} T^{2} ) \)
23$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 + 110 T + p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 520 T + p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 182 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \)
67$C_2$ \( ( 1 + 880 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 1190 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 - 884 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \)
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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

−17.63527759404419863783128520334, −17.38982325996438439646512944928, −16.34500091158247461705829394231, −16.30485985708187478834088296120, −15.52454943668811818083084724822, −15.11259072625038686657331790189, −14.23362462005558988849310461140, −13.53575954062218435451689079563, −12.37696182031449400309338652648, −12.16007576358184998281479011544, −11.30499475818402045982129225752, −10.77166038856977473910419787828, −10.00703873359342517455136202866, −9.084401795919869858826569941069, −7.83476027603686869972545400194, −7.24623828832099096945840415539, −6.09572951414993823770614548241, −5.89454847868845187990304039147, −3.48417398160779976635838678149, −2.39575046885036427936358363073, 2.39575046885036427936358363073, 3.48417398160779976635838678149, 5.89454847868845187990304039147, 6.09572951414993823770614548241, 7.24623828832099096945840415539, 7.83476027603686869972545400194, 9.084401795919869858826569941069, 10.00703873359342517455136202866, 10.77166038856977473910419787828, 11.30499475818402045982129225752, 12.16007576358184998281479011544, 12.37696182031449400309338652648, 13.53575954062218435451689079563, 14.23362462005558988849310461140, 15.11259072625038686657331790189, 15.52454943668811818083084724822, 16.30485985708187478834088296120, 16.34500091158247461705829394231, 17.38982325996438439646512944928, 17.63527759404419863783128520334

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