# Properties

 Label 4-2347884-1.1-c1e2-0-5 Degree $4$ Conductor $2347884$ Sign $-1$ Analytic cond. $149.703$ Root an. cond. $3.49790$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2·3-s − 4-s − 5-s + 3·9-s + 11-s + 2·12-s + 2·15-s + 16-s + 20-s − 7·23-s − 3·25-s − 4·27-s + 31-s − 2·33-s − 3·36-s + 7·37-s − 44-s − 3·45-s + 13·47-s − 2·48-s + 49-s − 5·53-s − 55-s + 3·59-s − 2·60-s − 64-s + 12·67-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1/2·4-s − 0.447·5-s + 9-s + 0.301·11-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 0.223·20-s − 1.45·23-s − 3/5·25-s − 0.769·27-s + 0.179·31-s − 0.348·33-s − 1/2·36-s + 1.15·37-s − 0.150·44-s − 0.447·45-s + 1.89·47-s − 0.288·48-s + 1/7·49-s − 0.686·53-s − 0.134·55-s + 0.390·59-s − 0.258·60-s − 1/8·64-s + 1.46·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2347884$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{3}$$ Sign: $-1$ Analytic conductor: $$149.703$$ Root analytic conductor: $$3.49790$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{2347884} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 2347884,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_2$ $$1 + T^{2}$$
3$C_1$ $$( 1 + T )^{2}$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
11$C_1$ $$1 - T$$
good5$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 13 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$C_2^2$ $$1 + 19 T^{2} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} )$$
41$C_2^2$ $$1 - 29 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} )$$
61$C_2^2$ $$1 + 36 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
73$C_2^2$ $$1 + 30 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 15 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
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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

−7.52762712697587657263010644471, −7.00402164374025168755318548697, −6.54214085601683384907620106545, −6.10878976323756887844324767104, −5.77503041166670714267004163067, −5.43400243199392372637399321908, −4.87572181271043075163269651748, −4.28269410667252079460638359096, −4.13789819548286247267062318976, −3.71850034705211522774469739571, −2.93406378582761935887723097576, −2.27134608949972785456222357257, −1.54393518050178245150443581761, −0.791960728109073280055411640713, 0, 0.791960728109073280055411640713, 1.54393518050178245150443581761, 2.27134608949972785456222357257, 2.93406378582761935887723097576, 3.71850034705211522774469739571, 4.13789819548286247267062318976, 4.28269410667252079460638359096, 4.87572181271043075163269651748, 5.43400243199392372637399321908, 5.77503041166670714267004163067, 6.10878976323756887844324767104, 6.54214085601683384907620106545, 7.00402164374025168755318548697, 7.52762712697587657263010644471