Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 19·2-s + 162·3-s + 429·4-s − 1.25e3·5-s + 3.07e3·6-s − 1.18e4·7-s + 1.91e4·8-s + 1.96e4·9-s − 2.37e4·10-s + 3.54e4·11-s + 6.94e4·12-s + 1.43e5·13-s − 2.25e5·14-s − 2.02e5·15-s + 3.16e5·16-s + 3.85e5·17-s + 3.73e5·18-s − 4.03e5·19-s − 5.36e5·20-s − 1.92e6·21-s + 6.74e5·22-s + 2.23e5·23-s + 3.10e6·24-s + 1.17e6·25-s + 2.72e6·26-s + 2.12e6·27-s − 5.09e6·28-s + ⋯
L(s)  = 1  + 0.839·2-s + 1.15·3-s + 0.837·4-s − 0.894·5-s + 0.969·6-s − 1.86·7-s + 1.65·8-s + 9-s − 0.751·10-s + 0.730·11-s + 0.967·12-s + 1.39·13-s − 1.56·14-s − 1.03·15-s + 1.20·16-s + 1.11·17-s + 0.839·18-s − 0.709·19-s − 0.749·20-s − 2.15·21-s + 0.613·22-s + 0.166·23-s + 1.91·24-s + 3/5·25-s + 1.17·26-s + 0.769·27-s − 1.56·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(225\)    =    \(3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{15} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 225,\ (\ :9/2, 9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(5.22582\)
\(L(\frac12)\)  \(\approx\)  \(5.22582\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{4} T )^{2} \)
5$C_1$ \( ( 1 + p^{4} T )^{2} \)
good2$D_{4}$ \( 1 - 19 T - 17 p^{2} T^{2} - 19 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 + 1696 p T + 327218 p^{3} T^{2} + 1696 p^{10} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 - 35488 T + 526013014 T^{2} - 35488 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 11052 p T + 24105149134 T^{2} - 11052 p^{10} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 - 385156 T + 268539949078 T^{2} - 385156 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 + 403296 T + 684929514838 T^{2} + 403296 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 - 223704 T - 1375273107794 T^{2} - 223704 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 74572 T + 28833430018078 T^{2} + 74572 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 + 5027128 T + 52415931233342 T^{2} + 5027128 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 - 5373628 T + 231061724951934 T^{2} - 5373628 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 14211332 T + 443988635955862 T^{2} - 14211332 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 - 27748920 T + 1170232974699430 T^{2} - 27748920 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 - 95966440 T + 4320659216802910 T^{2} - 95966440 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 + 64305596 T + 6611083028543086 T^{2} + 64305596 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 - 187863136 T + 23071633420288438 T^{2} - 187863136 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 154080060 T + 23302683905802238 T^{2} - 154080060 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 - 33592376 T - 10819815556424362 T^{2} - 33592376 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 + 228270976 T + 45777616900481806 T^{2} + 228270976 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 33122316 T + 68371107952007926 T^{2} + 33122316 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 + 932406760 T + 453226630902929438 T^{2} + 932406760 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 - 207040152 T + 372783310330485238 T^{2} - 207040152 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 - 2522676 p T + 610925899926766678 T^{2} - 2522676 p^{10} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 387134596 T - 734969029248610362 T^{2} - 387134596 p^{9} T^{3} + p^{18} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.21074291978532356264856715795, −16.19540715748169142862627741942, −16.15918161978580588306669409584, −15.61423861807197255757040028166, −14.56252578855415426272620302290, −14.23478208490347068356780115683, −13.11378504823525760733677885669, −13.06114903251926765807882633844, −12.20877336711187005618390922722, −11.14125365410158502085590269814, −10.40107408811533908524260023946, −9.454364224001635162685792574033, −8.584576281800275813544804282475, −7.54293670634805485007182826298, −6.89779808833438444439512651130, −5.91994363575284975170327005585, −3.96891381680544607615331011090, −3.81141853109217920931067793814, −2.70407076350904942917171515428, −1.10701398784709352907941481180, 1.10701398784709352907941481180, 2.70407076350904942917171515428, 3.81141853109217920931067793814, 3.96891381680544607615331011090, 5.91994363575284975170327005585, 6.89779808833438444439512651130, 7.54293670634805485007182826298, 8.584576281800275813544804282475, 9.454364224001635162685792574033, 10.40107408811533908524260023946, 11.14125365410158502085590269814, 12.20877336711187005618390922722, 13.06114903251926765807882633844, 13.11378504823525760733677885669, 14.23478208490347068356780115683, 14.56252578855415426272620302290, 15.61423861807197255757040028166, 16.15918161978580588306669409584, 16.19540715748169142862627741942, 17.21074291978532356264856715795

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