Properties

Label 4-21e2-1.1-c5e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $11.3438$
Root an. cond. $1.83522$
Motivic weight $5$
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  + 2·2-s + 9·3-s + 32·4-s − 11·5-s + 18·6-s + 259·7-s + 184·8-s − 22·10-s − 269·11-s + 288·12-s − 616·13-s + 518·14-s − 99·15-s + 368·16-s − 1.89e3·17-s + 164·19-s − 352·20-s + 2.33e3·21-s − 538·22-s + 3.26e3·23-s + 1.65e3·24-s + 3.12e3·25-s − 1.23e3·26-s − 729·27-s + 8.28e3·28-s + 4.83e3·29-s − 198·30-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.577·3-s + 4-s − 0.196·5-s + 0.204·6-s + 1.99·7-s + 1.01·8-s − 0.0695·10-s − 0.670·11-s + 0.577·12-s − 1.01·13-s + 0.706·14-s − 0.113·15-s + 0.359·16-s − 1.59·17-s + 0.104·19-s − 0.196·20-s + 1.15·21-s − 0.236·22-s + 1.28·23-s + 0.586·24-s + 25-s − 0.357·26-s − 0.192·27-s + 1.99·28-s + 1.06·29-s − 0.0401·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(11.3438\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.406010245\)
\(L(\frac12)\) \(\approx\) \(3.406010245\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 - p^{2} T + p^{4} T^{2} \)
7$C_2$ \( 1 - 37 p T + p^{5} T^{2} \)
good2$C_2^2$ \( 1 - p T - 7 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4} \)
5$C_2^2$ \( 1 + 11 T - 3004 T^{2} + 11 p^{5} T^{3} + p^{10} T^{4} \)
11$C_2^2$ \( 1 + 269 T - 88690 T^{2} + 269 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2$ \( ( 1 + 308 T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 1896 T + 2174959 T^{2} + 1896 p^{5} T^{3} + p^{10} T^{4} \)
19$C_2^2$ \( 1 - 164 T - 2449203 T^{2} - 164 p^{5} T^{3} + p^{10} T^{4} \)
23$C_2^2$ \( 1 - 3264 T + 4217353 T^{2} - 3264 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 2417 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2841 T - 20557870 T^{2} + 2841 p^{5} T^{3} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 11328 T + 58979627 T^{2} - 11328 p^{5} T^{3} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 16856 T + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 7894 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 21102 T + 215949397 T^{2} + 21102 p^{5} T^{3} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 29691 T + 463359988 T^{2} - 29691 p^{5} T^{3} + p^{10} T^{4} \)
59$C_2^2$ \( 1 - 8163 T - 648289730 T^{2} - 8163 p^{5} T^{3} + p^{10} T^{4} \)
61$C_2^2$ \( 1 + 15166 T - 614588745 T^{2} + 15166 p^{5} T^{3} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 32078 T - 321127023 T^{2} - 32078 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 38274 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 34866 T - 857433637 T^{2} + 34866 p^{5} T^{3} + p^{10} T^{4} \)
79$C_2^2$ \( 1 + 13529 T - 2894022558 T^{2} + 13529 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2$ \( ( 1 + 68103 T + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114922 T + 7623006635 T^{2} - 114922 p^{5} T^{3} + p^{10} T^{4} \)
97$C_2$ \( ( 1 - 154959 T + p^{5} T^{2} )^{2} \)
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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

−17.07992254346787361207996444366, −17.05913613541998908204235982049, −15.96637152045561049916619260318, −15.34398538378447585864942778982, −14.67068236937261504725333614358, −14.64668832105816462542178637351, −13.32028728743162366899138785025, −13.28487970260665828998566639764, −11.69275279578636196646526949339, −11.68393884921847301776416933191, −10.78221884022388611520688050872, −10.26754123727480900395552315345, −8.754702276809893504435825252344, −8.270496920005879449165935584238, −7.35144283289989612574614320361, −6.81750658270653134090017068062, −4.97554966483568235792382431637, −4.69903887003035454673668142909, −2.72825060340540242553584079017, −1.75412286179518563771849019185, 1.75412286179518563771849019185, 2.72825060340540242553584079017, 4.69903887003035454673668142909, 4.97554966483568235792382431637, 6.81750658270653134090017068062, 7.35144283289989612574614320361, 8.270496920005879449165935584238, 8.754702276809893504435825252344, 10.26754123727480900395552315345, 10.78221884022388611520688050872, 11.68393884921847301776416933191, 11.69275279578636196646526949339, 13.28487970260665828998566639764, 13.32028728743162366899138785025, 14.64668832105816462542178637351, 14.67068236937261504725333614358, 15.34398538378447585864942778982, 15.96637152045561049916619260318, 17.05913613541998908204235982049, 17.07992254346787361207996444366

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