Properties

Label 4-147e2-1.1-c5e2-0-2
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $555.847$
Root an. cond. $4.85555$
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  + 6·2-s + 9·3-s + 32·4-s − 78·5-s + 54·6-s + 360·8-s − 468·10-s − 444·11-s + 288·12-s − 884·13-s − 702·15-s + 2.16e3·16-s + 126·17-s − 2.68e3·19-s − 2.49e3·20-s − 2.66e3·22-s − 4.20e3·23-s + 3.24e3·24-s + 3.12e3·25-s − 5.30e3·26-s − 729·27-s − 1.08e4·29-s − 4.21e3·30-s − 80·31-s + 1.15e4·32-s − 3.99e3·33-s + 756·34-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 4-s − 1.39·5-s + 0.612·6-s + 1.98·8-s − 1.47·10-s − 1.10·11-s + 0.577·12-s − 1.45·13-s − 0.805·15-s + 2.10·16-s + 0.105·17-s − 1.70·19-s − 1.39·20-s − 1.17·22-s − 1.65·23-s + 1.14·24-s + 25-s − 1.53·26-s − 0.192·27-s − 2.40·29-s − 0.854·30-s − 0.0149·31-s + 1.98·32-s − 0.638·33-s + 0.112·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(555.847\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.134816897\)
\(L(\frac12)\) \(\approx\) \(2.134816897\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 - 3 p T + p^{2} T^{2} - 3 p^{6} T^{3} + p^{10} T^{4} \)
5$C_2^2$ \( 1 + 78 T + 2959 T^{2} + 78 p^{5} T^{3} + p^{10} T^{4} \)
11$C_2^2$ \( 1 + 444 T + 36085 T^{2} + 444 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2$ \( ( 1 + 34 p T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 126 T - 1403981 T^{2} - 126 p^{5} T^{3} + p^{10} T^{4} \)
19$C_2^2$ \( 1 + 2684 T + 4727757 T^{2} + 2684 p^{5} T^{3} + p^{10} T^{4} \)
23$C_2^2$ \( 1 + 4200 T + 11203657 T^{2} + 4200 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 5442 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 80 T - 28622751 T^{2} + 80 p^{5} T^{3} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 5434 T - 39815601 T^{2} - 5434 p^{5} T^{3} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 7962 T + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 268 p T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 13920 T - 35578607 T^{2} - 13920 p^{5} T^{3} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 9594 T - 326150657 T^{2} - 9594 p^{5} T^{3} + p^{10} T^{4} \)
59$C_2^2$ \( 1 + 27492 T + 40885765 T^{2} + 27492 p^{5} T^{3} + p^{10} T^{4} \)
61$C_2^2$ \( 1 + 49478 T + 1603476183 T^{2} + 49478 p^{5} T^{3} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 59356 T + 2173009629 T^{2} - 59356 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 32040 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 61846 T + 1751856123 T^{2} - 61846 p^{5} T^{3} + p^{10} T^{4} \)
79$C_2^2$ \( 1 - 65776 T + 1249425777 T^{2} - 65776 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2$ \( ( 1 - 40188 T + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 7974 T - 5520474773 T^{2} - 7974 p^{5} T^{3} + p^{10} T^{4} \)
97$C_2$ \( ( 1 + 143662 T + p^{5} T^{2} )^{2} \)
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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

−12.37267942221707701866710971737, −12.25735097790308652579408349638, −11.33722347091717048290249089610, −10.92813965320586952857455138305, −10.70787242846637690066120869274, −9.866477786776292022879285271579, −9.504787738818300343770048651040, −8.268769414663246479157885117185, −8.110807781561089577122431050035, −7.52306568955418276409634218132, −7.37058665093314473129618239347, −6.50131624677641011352285160093, −5.67061899817867306119845861452, −5.00084021470008397935389571515, −4.39714423639403199621530903871, −3.97110970669868766338939510135, −3.35684217607051297998196800219, −2.18573393359572254728069463825, −2.12192349625375215475327525950, −0.37103823368060179196664231475, 0.37103823368060179196664231475, 2.12192349625375215475327525950, 2.18573393359572254728069463825, 3.35684217607051297998196800219, 3.97110970669868766338939510135, 4.39714423639403199621530903871, 5.00084021470008397935389571515, 5.67061899817867306119845861452, 6.50131624677641011352285160093, 7.37058665093314473129618239347, 7.52306568955418276409634218132, 8.110807781561089577122431050035, 8.268769414663246479157885117185, 9.504787738818300343770048651040, 9.866477786776292022879285271579, 10.70787242846637690066120869274, 10.92813965320586952857455138305, 11.33722347091717048290249089610, 12.25735097790308652579408349638, 12.37267942221707701866710971737

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