# Properties

 Label 4-35e2-1.1-c9e2-0-0 Degree $4$ Conductor $1225$ Sign $1$ Analytic cond. $324.945$ Root an. cond. $4.24573$ Motivic weight $9$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 24·2-s − 174·3-s − 584·4-s + 1.25e3·5-s + 4.17e3·6-s + 4.80e3·7-s + 2.95e4·8-s + 6.66e3·9-s − 3.00e4·10-s + 1.85e4·11-s + 1.01e5·12-s − 5.10e4·13-s − 1.15e5·14-s − 2.17e5·15-s + 576·16-s − 3.73e5·17-s − 1.60e5·18-s − 1.43e5·19-s − 7.30e5·20-s − 8.35e5·21-s − 4.45e5·22-s − 4.98e5·23-s − 5.14e6·24-s + 1.17e6·25-s + 1.22e6·26-s − 4.77e5·27-s − 2.80e6·28-s + ⋯
 L(s)  = 1 − 1.06·2-s − 1.24·3-s − 1.14·4-s + 0.894·5-s + 1.31·6-s + 0.755·7-s + 2.55·8-s + 0.338·9-s − 0.948·10-s + 0.382·11-s + 1.41·12-s − 0.496·13-s − 0.801·14-s − 1.10·15-s + 0.00219·16-s − 1.08·17-s − 0.359·18-s − 0.252·19-s − 1.02·20-s − 0.937·21-s − 0.405·22-s − 0.371·23-s − 3.16·24-s + 3/5·25-s + 0.526·26-s − 0.172·27-s − 0.862·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$324.945$$ Root analytic conductor: $$4.24573$$ Motivic weight: $$9$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1225,\ (\ :9/2, 9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ 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)$
bad5$C_1$ $$( 1 - p^{4} T )^{2}$$
7$C_1$ $$( 1 - p^{4} T )^{2}$$
good2$D_{4}$ $$1 + 3 p^{3} T + 145 p^{3} T^{2} + 3 p^{12} T^{3} + p^{18} T^{4}$$
3$D_{4}$ $$1 + 58 p T + 2623 p^{2} T^{2} + 58 p^{10} T^{3} + p^{18} T^{4}$$
11$D_{4}$ $$1 - 18566 T + 4136090463 T^{2} - 18566 p^{9} T^{3} + p^{18} T^{4}$$
13$D_{4}$ $$1 + 3930 p T + 19172461323 T^{2} + 3930 p^{10} T^{3} + p^{18} T^{4}$$
17$D_{4}$ $$1 + 373910 T + 270283008251 T^{2} + 373910 p^{9} T^{3} + p^{18} T^{4}$$
19$D_{4}$ $$1 + 143276 T + 642750798250 T^{2} + 143276 p^{9} T^{3} + p^{18} T^{4}$$
23$D_{4}$ $$1 + 498908 T - 1733603053870 T^{2} + 498908 p^{9} T^{3} + p^{18} T^{4}$$
29$D_{4}$ $$1 + 399226 p T + 61819642147595 T^{2} + 399226 p^{10} T^{3} + p^{18} T^{4}$$
31$D_{4}$ $$1 + 3953760 T + 18380996896542 T^{2} + 3953760 p^{9} T^{3} + p^{18} T^{4}$$
37$D_{4}$ $$1 + 3205412 T + 134679597433390 T^{2} + 3205412 p^{9} T^{3} + p^{18} T^{4}$$
41$D_{4}$ $$1 - 1058992 T - 202350146478094 T^{2} - 1058992 p^{9} T^{3} + p^{18} T^{4}$$
43$D_{4}$ $$1 - 15948180 T + 312060834724314 T^{2} - 15948180 p^{9} T^{3} + p^{18} T^{4}$$
47$D_{4}$ $$1 - 65501290 T + 2591944660543287 T^{2} - 65501290 p^{9} T^{3} + p^{18} T^{4}$$
53$D_{4}$ $$1 + 25114688 T + 6536128371514410 T^{2} + 25114688 p^{9} T^{3} + p^{18} T^{4}$$
59$D_{4}$ $$1 + 116159208 T + 8473255943386694 T^{2} + 116159208 p^{9} T^{3} + p^{18} T^{4}$$
61$D_{4}$ $$1 + 44688544 T + 21891928402164378 T^{2} + 44688544 p^{9} T^{3} + p^{18} T^{4}$$
67$D_{4}$ $$1 - 118092496 T + 11543378662237830 T^{2} - 118092496 p^{9} T^{3} + p^{18} T^{4}$$
71$D_{4}$ $$1 + 294165824 T + 66419314231297806 T^{2} + 294165824 p^{9} T^{3} + p^{18} T^{4}$$
73$D_{4}$ $$1 + 57419332 T + 116103868936695574 T^{2} + 57419332 p^{9} T^{3} + p^{18} T^{4}$$
79$D_{4}$ $$1 + 692852854 T + 353229306838520119 T^{2} + 692852854 p^{9} T^{3} + p^{18} T^{4}$$
83$D_{4}$ $$1 + 6514216 p T + 378894185867322102 T^{2} + 6514216 p^{10} T^{3} + p^{18} T^{4}$$
89$D_{4}$ $$1 + 779043704 T + 776620283810146850 T^{2} + 779043704 p^{9} T^{3} + p^{18} T^{4}$$
97$D_{4}$ $$1 + 2673039406 T + 3290052660205451443 T^{2} + 2673039406 p^{9} T^{3} + p^{18} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$