Properties

Label 4-21e2-1.1-c3e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·3-s + 5·4-s + 6·5-s − 18·6-s + 14·7-s − 27·8-s + 27·9-s − 18·10-s − 6·11-s + 30·12-s + 16·13-s − 42·14-s + 36·15-s + 69·16-s − 6·17-s − 81·18-s + 64·19-s + 30·20-s + 84·21-s + 18·22-s + 6·23-s − 162·24-s − 166·25-s − 48·26-s + 108·27-s + 70·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 1.15·3-s + 5/8·4-s + 0.536·5-s − 1.22·6-s + 0.755·7-s − 1.19·8-s + 9-s − 0.569·10-s − 0.164·11-s + 0.721·12-s + 0.341·13-s − 0.801·14-s + 0.619·15-s + 1.07·16-s − 0.0856·17-s − 1.06·18-s + 0.772·19-s + 0.335·20-s + 0.872·21-s + 0.174·22-s + 0.0543·23-s − 1.37·24-s − 1.32·25-s − 0.362·26-s + 0.769·27-s + 0.472·28-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

\(L(2)\) \(\approx\) \(1.057700512\)
\(L(\frac12)\) \(\approx\) \(1.057700512\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 + 3 T + p^{2} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 6 T + 202 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 16 T + 2406 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \)
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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

−17.98754763658970438917881986449, −17.57422945386022424369784705375, −17.04459096373884350284302281883, −15.98086122027751495503143187930, −15.48248129646803441342299119708, −14.95781607476672701284447506651, −14.11173331179018455937931545626, −13.73575929130420354671972248254, −12.86151962394381771607778953262, −11.95403214843741524545736968147, −11.21421215607790565009657466779, −10.29660072704947938309086752171, −9.382639145802876843343958417977, −9.243114880363181102766207436345, −8.177937000406163814747537789084, −7.77893703428859490642609174030, −6.57897767162756355612751119591, −5.34722196807412904954605676407, −3.52383899701414130274827904738, −1.97603540068635921626971898304, 1.97603540068635921626971898304, 3.52383899701414130274827904738, 5.34722196807412904954605676407, 6.57897767162756355612751119591, 7.77893703428859490642609174030, 8.177937000406163814747537789084, 9.243114880363181102766207436345, 9.382639145802876843343958417977, 10.29660072704947938309086752171, 11.21421215607790565009657466779, 11.95403214843741524545736968147, 12.86151962394381771607778953262, 13.73575929130420354671972248254, 14.11173331179018455937931545626, 14.95781607476672701284447506651, 15.48248129646803441342299119708, 15.98086122027751495503143187930, 17.04459096373884350284302281883, 17.57422945386022424369784705375, 17.98754763658970438917881986449

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