# Properties

 Label 4-1815e2-1.1-c1e2-0-28 Degree $4$ Conductor $3294225$ Sign $-1$ Analytic cond. $210.042$ Root an. cond. $3.80694$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + 3-s + 5-s − 2·9-s + 15-s − 4·16-s + 2·23-s − 4·25-s − 5·27-s + 14·31-s − 2·45-s + 16·47-s − 4·48-s − 10·49-s − 12·53-s + 2·69-s − 4·75-s − 4·80-s + 81-s + 14·93-s − 18·113-s + 2·115-s − 9·125-s + 127-s + 131-s − 5·135-s + 137-s + 139-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.258·15-s − 16-s + 0.417·23-s − 4/5·25-s − 0.962·27-s + 2.51·31-s − 0.298·45-s + 2.33·47-s − 0.577·48-s − 1.42·49-s − 1.64·53-s + 0.240·69-s − 0.461·75-s − 0.447·80-s + 1/9·81-s + 1.45·93-s − 1.69·113-s + 0.186·115-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s − 0.430·135-s + 0.0854·137-s + 0.0848·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3294225$$    =    $$3^{2} \cdot 5^{2} \cdot 11^{4}$$ Sign: $-1$ Analytic conductor: $$210.042$$ Root analytic conductor: $$3.80694$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{3294225} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 3294225,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 - T + p T^{2}$$
5$C_2$ $$1 - T + p T^{2}$$
11 $$1$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
19$C_2$ $$( 1 - p T^{2} )^{2}$$
23$C_2$ $$( 1 - T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
41$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
71$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
73$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
79$C_2^2$ $$1 - 58 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
97$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
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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

−7.39893594642771643206543547031, −6.80044609023638042209450417099, −6.47231251560475429244143852591, −6.09966792552373038662002103453, −5.73870626081545475442824548628, −5.13615482341982682420909731232, −4.77016036292615901681122983244, −4.30413573577731527020603612370, −3.82123745760857088121774501968, −3.21993795263974952405435891110, −2.67401217228157830548156869283, −2.46478514923094428626030938829, −1.78907311686180992193673629119, −1.03757175929891373992477223583, 0, 1.03757175929891373992477223583, 1.78907311686180992193673629119, 2.46478514923094428626030938829, 2.67401217228157830548156869283, 3.21993795263974952405435891110, 3.82123745760857088121774501968, 4.30413573577731527020603612370, 4.77016036292615901681122983244, 5.13615482341982682420909731232, 5.73870626081545475442824548628, 6.09966792552373038662002103453, 6.47231251560475429244143852591, 6.80044609023638042209450417099, 7.39893594642771643206543547031