# Properties

 Label 4-739328-1.1-c1e2-0-7 Degree $4$ Conductor $739328$ Sign $-1$ Analytic cond. $47.1401$ Root an. cond. $2.62028$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 5·9-s + 10·17-s + 6·25-s − 24·41-s − 13·49-s − 14·73-s + 16·81-s − 12·89-s + 12·97-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 50·153-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 5/3·9-s + 2.42·17-s + 6/5·25-s − 3.74·41-s − 1.85·49-s − 1.63·73-s + 16/9·81-s − 1.27·89-s + 1.21·97-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 4.04·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$739328$$    =    $$2^{11} \cdot 19^{2}$$ Sign: $-1$ Analytic conductor: $$47.1401$$ Root analytic conductor: $$2.62028$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{739328} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 739328,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad2 $$1$$
19$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2^2$ $$1 - 25 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
29$C_2^2$ $$1 - 33 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2^2$ $$1 - 25 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
61$C_2^2$ $$1 - 118 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
71$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
73$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
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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

−8.097977130148826306537543729545, −7.78270604734653611245586904422, −7.17256961441030457549618001314, −6.66497413006612600364626607654, −6.30023261682060838291785304978, −5.64331877998247196033481429906, −5.35462021395561592211452786575, −5.05471255286984233114047950550, −4.38790175941963038428793452679, −3.39826650555000626660333262631, −3.25408706841441563001400294407, −2.91676218961444217862223110393, −1.89564027396146231703355154113, −1.17643257441768604939588188061, 0, 1.17643257441768604939588188061, 1.89564027396146231703355154113, 2.91676218961444217862223110393, 3.25408706841441563001400294407, 3.39826650555000626660333262631, 4.38790175941963038428793452679, 5.05471255286984233114047950550, 5.35462021395561592211452786575, 5.64331877998247196033481429906, 6.30023261682060838291785304978, 6.66497413006612600364626607654, 7.17256961441030457549618001314, 7.78270604734653611245586904422, 8.097977130148826306537543729545