Properties

Label 4-21e2-1.1-c13e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $507.082$
Root an. cond. $4.74536$
Motivic weight $13$
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  + 144·2-s + 1.45e3·3-s + 1.52e3·4-s − 5.25e4·5-s + 2.09e5·6-s + 2.35e5·7-s − 1.36e6·8-s + 1.59e6·9-s − 7.56e6·10-s − 8.59e6·11-s + 2.21e6·12-s − 3.79e7·13-s + 3.38e7·14-s − 7.66e7·15-s − 1.23e8·16-s + 1.66e7·17-s + 2.29e8·18-s − 2.11e8·19-s − 7.98e7·20-s + 3.43e8·21-s − 1.23e9·22-s − 1.94e8·23-s − 1.99e9·24-s + 9.18e8·25-s − 5.46e9·26-s + 1.54e9·27-s + 3.57e8·28-s + ⋯
L(s)  = 1  + 1.59·2-s + 1.15·3-s + 0.185·4-s − 1.50·5-s + 1.83·6-s + 0.755·7-s − 1.84·8-s + 9-s − 2.39·10-s − 1.46·11-s + 0.214·12-s − 2.18·13-s + 1.20·14-s − 1.73·15-s − 1.84·16-s + 0.167·17-s + 1.59·18-s − 1.02·19-s − 0.279·20-s + 0.872·21-s − 2.32·22-s − 0.274·23-s − 2.13·24-s + 0.752·25-s − 3.47·26-s + 0.769·27-s + 0.140·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(507.082\)
Root analytic conductor: \(4.74536\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 441,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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$C_1$ \( ( 1 - p^{6} T )^{2} \)
7$C_1$ \( ( 1 - p^{6} T )^{2} \)
good2$D_{4}$ \( 1 - 9 p^{4} T + 1201 p^{4} T^{2} - 9 p^{17} T^{3} + p^{26} T^{4} \)
5$D_{4}$ \( 1 + 10512 p T + 73763578 p^{2} T^{2} + 10512 p^{14} T^{3} + p^{26} T^{4} \)
11$D_{4}$ \( 1 + 781236 p T + 551670639946 p^{2} T^{2} + 781236 p^{14} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 + 37982012 T + 854172190564734 T^{2} + 37982012 p^{13} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 - 16643088 T + 4110551987575522 T^{2} - 16643088 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 211195304 T + 37306293658457814 T^{2} + 211195304 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 + 194997852 T + 640913258073572530 T^{2} + 194997852 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 + 3481642548 T + 6322342576139078206 T^{2} + 3481642548 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 + 8393797520 T + 47374586993368068990 T^{2} + 8393797520 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 + 22337946716 T + \)\(56\!\cdots\!86\)\( T^{2} + 22337946716 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 + 12507403320 T + \)\(12\!\cdots\!42\)\( T^{2} + 12507403320 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 - 26062746328 T + \)\(32\!\cdots\!54\)\( T^{2} - 26062746328 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 - 29222658936 T + \)\(49\!\cdots\!30\)\( T^{2} - 29222658936 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 - 523911868308 T + \)\(12\!\cdots\!90\)\( T^{2} - 523911868308 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 + 94444581024 T + \)\(21\!\cdots\!14\)\( T^{2} + 94444581024 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 - 461720554252 T + \)\(31\!\cdots\!26\)\( T^{2} - 461720554252 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 + 1237879942064 T + \)\(14\!\cdots\!30\)\( T^{2} + 1237879942064 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 - 1961126619804 T + \)\(32\!\cdots\!26\)\( T^{2} - 1961126619804 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 + 2407791783284 T + \)\(47\!\cdots\!62\)\( T^{2} + 2407791783284 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 + 2622494223848 T + \)\(11\!\cdots\!42\)\( T^{2} + 2622494223848 p^{13} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 - 4489561815912 T + \)\(20\!\cdots\!90\)\( T^{2} - 4489561815912 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 - 520927840584 T + \)\(43\!\cdots\!50\)\( T^{2} - 520927840584 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 + 13027956631604 T + \)\(17\!\cdots\!86\)\( T^{2} + 13027956631604 p^{13} T^{3} + p^{26} T^{4} \)
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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

−14.65590751716431028639040140083, −14.26227205068090053689335450945, −13.35409238367345895075591483821, −12.98989220503368054268139818674, −12.19894098090498121735117302799, −12.02191579399852706233448793998, −10.80780546020790169533284303884, −9.999282118151893424843076686468, −8.966425352135372656037529158593, −8.426708799007276381686177857521, −7.49204946915743827333780400013, −7.34711936090113565869909731791, −5.34999442661610907200652220046, −5.09984952624912156157940130492, −4.00418300858214072081370990258, −3.96114119051193182764480743006, −2.79404467674201695885620491172, −2.16139889903251882746718150007, 0, 0, 2.16139889903251882746718150007, 2.79404467674201695885620491172, 3.96114119051193182764480743006, 4.00418300858214072081370990258, 5.09984952624912156157940130492, 5.34999442661610907200652220046, 7.34711936090113565869909731791, 7.49204946915743827333780400013, 8.426708799007276381686177857521, 8.966425352135372656037529158593, 9.999282118151893424843076686468, 10.80780546020790169533284303884, 12.02191579399852706233448793998, 12.19894098090498121735117302799, 12.98989220503368054268139818674, 13.35409238367345895075591483821, 14.26227205068090053689335450945, 14.65590751716431028639040140083

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