Properties

Label 4-21e2-1.1-c4e2-0-2
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $4.71223$
Root an. cond. $1.47335$
Motivic weight $4$
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  + 5·2-s + 9·3-s + 16·4-s − 3·5-s + 45·6-s − 91·7-s + 115·8-s + 54·9-s − 15·10-s + 149·11-s + 144·12-s − 455·14-s − 27·15-s + 575·16-s − 462·17-s + 270·18-s − 618·19-s − 48·20-s − 819·21-s + 745·22-s + 560·23-s + 1.03e3·24-s − 619·25-s + 243·27-s − 1.45e3·28-s + 470·29-s − 135·30-s + ⋯
L(s)  = 1  + 5/4·2-s + 3-s + 4-s − 0.119·5-s + 5/4·6-s − 1.85·7-s + 1.79·8-s + 2/3·9-s − 0.149·10-s + 1.23·11-s + 12-s − 2.32·14-s − 0.119·15-s + 2.24·16-s − 1.59·17-s + 5/6·18-s − 1.71·19-s − 0.119·20-s − 1.85·21-s + 1.53·22-s + 1.05·23-s + 1.79·24-s − 0.990·25-s + 1/3·27-s − 1.85·28-s + 0.558·29-s − 0.149·30-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.71223\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.452132357\)
\(L(\frac12)\) \(\approx\) \(3.452132357\)
\(L(3)\) 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$C_2$ \( 1 - p^{2} T + p^{3} T^{2} \)
7$C_2$ \( 1 + 13 p T + p^{4} T^{2} \)
good2$C_2^2$ \( 1 - 5 T + 9 T^{2} - 5 p^{4} T^{3} + p^{8} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 628 T^{2} + 3 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2^2$ \( 1 - 149 T + 7560 T^{2} - 149 p^{4} T^{3} + p^{8} T^{4} \)
13$C_2^2$ \( 1 - 55394 T^{2} + p^{8} T^{4} \)
17$C_2^2$ \( 1 + 462 T + 154669 T^{2} + 462 p^{4} T^{3} + p^{8} T^{4} \)
19$C_2^2$ \( 1 + 618 T + 257629 T^{2} + 618 p^{4} T^{3} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 560 T + 33759 T^{2} - 560 p^{4} T^{3} + p^{8} T^{4} \)
29$C_2$ \( ( 1 - 235 T + p^{4} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2229 T + 2579668 T^{2} + 2229 p^{4} T^{3} + p^{8} T^{4} \)
37$C_2^2$ \( 1 + 1970 T + 2006739 T^{2} + 1970 p^{4} T^{3} + p^{8} T^{4} \)
41$C_2^2$ \( 1 + 2280106 T^{2} + p^{8} T^{4} \)
43$C_2$ \( ( 1 - 2798 T + p^{4} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 7152 T + 21930049 T^{2} - 7152 p^{4} T^{3} + p^{8} T^{4} \)
53$C_2^2$ \( 1 + 17 p T - 2520 p^{2} T^{2} + 17 p^{5} T^{3} + p^{8} T^{4} \)
59$C_2^2$ \( 1 - 2391 T + 14022988 T^{2} - 2391 p^{4} T^{3} + p^{8} T^{4} \)
61$C_2^2$ \( 1 + 6180 T + 26576641 T^{2} + 6180 p^{4} T^{3} + p^{8} T^{4} \)
67$C_2^2$ \( 1 - 4156 T - 2878785 T^{2} - 4156 p^{4} T^{3} + p^{8} T^{4} \)
71$C_2$ \( ( 1 - 484 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1644 T + 29299153 T^{2} + 1644 p^{4} T^{3} + p^{8} T^{4} \)
79$C_2^2$ \( 1 + 4325 T - 20244456 T^{2} + 4325 p^{4} T^{3} + p^{8} T^{4} \)
83$C_2^2$ \( 1 - 93215615 T^{2} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 3426 T + 66654733 T^{2} - 3426 p^{4} T^{3} + p^{8} T^{4} \)
97$C_2^2$ \( 1 - 147347735 T^{2} + p^{8} T^{4} \)
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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

−17.26071596033061159917811024032, −17.06353876396204951004320823079, −16.14672090410641187392522148220, −15.61037131754701026876493797503, −15.17891062931089409339460142661, −14.19048938470903591684688158815, −13.96373222339278553526734980912, −13.08901447454589575230017439325, −12.84236913370184628840375047930, −12.23533153727057564348312320574, −10.85902140365596966638594164479, −10.53789861929356622909581032783, −9.111807104244553977633501411992, −9.017840311828740369852225580106, −7.35567519262057096249294130785, −6.85229238609290911030047716372, −5.91502003178006900602282358491, −4.15887196620337924832531694878, −3.83444810210489406457311791242, −2.32499222925034976875112394581, 2.32499222925034976875112394581, 3.83444810210489406457311791242, 4.15887196620337924832531694878, 5.91502003178006900602282358491, 6.85229238609290911030047716372, 7.35567519262057096249294130785, 9.017840311828740369852225580106, 9.111807104244553977633501411992, 10.53789861929356622909581032783, 10.85902140365596966638594164479, 12.23533153727057564348312320574, 12.84236913370184628840375047930, 13.08901447454589575230017439325, 13.96373222339278553526734980912, 14.19048938470903591684688158815, 15.17891062931089409339460142661, 15.61037131754701026876493797503, 16.14672090410641187392522148220, 17.06353876396204951004320823079, 17.26071596033061159917811024032

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