# Properties

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

# Learn more about

## 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$D_{4}$ $$1 - 387134596 T - 734969029248610362 T^{2} - 387134596 p^{9} T^{3} + p^{18} T^{4}$$
show more
show less
\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