Properties

Label 4-21e2-1.1-c9e2-0-2
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $116.980$
Root an. cond. $3.28873$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s − 162·3-s − 375·4-s + 1.17e3·5-s − 1.45e3·6-s − 4.80e3·7-s − 2.87e3·8-s + 1.96e4·9-s + 1.05e4·10-s − 1.45e5·11-s + 6.07e4·12-s + 8.65e4·13-s − 4.32e4·14-s − 1.89e5·15-s − 7.55e4·16-s − 2.29e5·17-s + 1.77e5·18-s − 2.21e5·19-s − 4.38e5·20-s + 7.77e5·21-s − 1.31e6·22-s − 2.03e6·23-s + 4.65e5·24-s + 2.85e3·25-s + 7.78e5·26-s − 2.12e6·27-s + 1.80e6·28-s + ⋯
L(s)  = 1  + 0.397·2-s − 1.15·3-s − 0.732·4-s + 0.837·5-s − 0.459·6-s − 0.755·7-s − 0.247·8-s + 9-s + 0.332·10-s − 3.00·11-s + 0.845·12-s + 0.840·13-s − 0.300·14-s − 0.966·15-s − 0.288·16-s − 0.667·17-s + 0.397·18-s − 0.389·19-s − 0.613·20-s + 0.872·21-s − 1.19·22-s − 1.51·23-s + 0.286·24-s + 0.00145·25-s + 0.334·26-s − 0.769·27-s + 0.553·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(116.980\)
Root analytic conductor: \(3.28873\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 441,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{2} \)
7$C_1$ \( ( 1 + p^{4} T )^{2} \)
good2$D_{4}$ \( 1 - 9 T + 57 p^{3} T^{2} - 9 p^{9} T^{3} + p^{18} T^{4} \)
5$D_{4}$ \( 1 - 234 p T + 54642 p^{2} T^{2} - 234 p^{10} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 + 145746 T + 9848423886 T^{2} + 145746 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 512 p^{2} T + 1774916238 p T^{2} - 512 p^{11} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 + 229842 T - 51722753798 T^{2} + 229842 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 + 221224 T + 34608506226 p T^{2} + 221224 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 + 2035782 T + 4618931326270 T^{2} + 2035782 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 - 9756252 T + 52470669069246 T^{2} - 9756252 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 - 204000 T + 30790423515710 T^{2} - 204000 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 + 13959816 T + 306736999959926 T^{2} + 13959816 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 + 42362550 T + 1063812640967922 T^{2} + 42362550 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 + 4763912 T + 345063686509734 T^{2} + 4763912 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 + 48278484 T + 2529195823681630 T^{2} + 48278484 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 + 108980352 T + 6796561315809670 T^{2} + 108980352 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 + 188376804 T + 23346971643078934 T^{2} + 188376804 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 19722092 T + 14767573710118686 T^{2} - 19722092 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 - 70274396 T - 15240809102766810 T^{2} - 70274396 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 + 382044186 T + 120425517984233086 T^{2} + 382044186 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 - 191785896 T + 124167221244272702 T^{2} - 191785896 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 + 72592148 T + 239185581484447902 T^{2} + 72592148 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 - 187994232 T + 258704825037511030 T^{2} - 187994232 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 - 42930954 T + 403096805893580370 T^{2} - 42930954 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 1726854096 T + 1929285565356657566 T^{2} - 1726854096 p^{9} T^{3} + p^{18} T^{4} \)
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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

−15.83139162482252357473044876303, −15.49541559287599524310183021086, −13.89687928361793056766950302852, −13.68902968831358412376817876056, −13.05545922290603916651832206709, −12.71715990572954696256751932006, −11.82839392153094824020859239329, −10.71973740563353815841164444833, −10.29582012035279688001517918774, −9.924445874878481394860658165431, −8.651743361050310897874362758882, −7.936439085720018069700667637154, −6.57518847137249302850107887911, −6.05670589295717238459541032237, −5.05585035875809063093404698337, −4.74382019086367707716020610926, −3.17569176210837533125277178296, −1.93473800591869434364385179935, 0, 0, 1.93473800591869434364385179935, 3.17569176210837533125277178296, 4.74382019086367707716020610926, 5.05585035875809063093404698337, 6.05670589295717238459541032237, 6.57518847137249302850107887911, 7.936439085720018069700667637154, 8.651743361050310897874362758882, 9.924445874878481394860658165431, 10.29582012035279688001517918774, 10.71973740563353815841164444833, 11.82839392153094824020859239329, 12.71715990572954696256751932006, 13.05545922290603916651832206709, 13.68902968831358412376817876056, 13.89687928361793056766950302852, 15.49541559287599524310183021086, 15.83139162482252357473044876303

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