# Properties

 Label 4-5290e2-1.1-c1e2-0-10 Degree $4$ Conductor $27984100$ Sign $1$ Analytic cond. $1784.29$ Root an. cond. $6.49929$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

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## Dirichlet series

 L(s)  = 1 + 2·2-s + 3-s + 3·4-s − 2·5-s + 2·6-s − 7-s + 4·8-s − 4·9-s − 4·10-s − 11-s + 3·12-s − 3·13-s − 2·14-s − 2·15-s + 5·16-s − 17-s − 8·18-s + 3·19-s − 6·20-s − 21-s − 2·22-s + 4·24-s + 3·25-s − 6·26-s − 6·27-s − 3·28-s − 14·29-s + ⋯
 L(s)  = 1 + 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s + 1.41·8-s − 4/3·9-s − 1.26·10-s − 0.301·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.516·15-s + 5/4·16-s − 0.242·17-s − 1.88·18-s + 0.688·19-s − 1.34·20-s − 0.218·21-s − 0.426·22-s + 0.816·24-s + 3/5·25-s − 1.17·26-s − 1.15·27-s − 0.566·28-s − 2.59·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$27984100$$    =    $$2^{2} \cdot 5^{2} \cdot 23^{4}$$ Sign: $1$ Analytic conductor: $$1784.29$$ Root analytic conductor: $$6.49929$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 27984100,\ (\ :1/2, 1/2),\ 1)$$

## 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$C_1$ $$( 1 - T )^{2}$$
5$C_1$ $$( 1 + T )^{2}$$
23 $$1$$
good3$D_{4}$ $$1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + T + p T^{2} + p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$D_{4}$ $$1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
89$C_4$ $$1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 27 T + 375 T^{2} + 27 p T^{3} + p^{2} T^{4}$$
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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

−7.81906273187618179676538576740, −7.75274728439861315120651588333, −7.08648667788623873489791526150, −7.08270744356990329242129282345, −6.56558165991719062802854989718, −6.15620996062020129395201056662, −5.54381359289116914041628820766, −5.52344065907571606240257717444, −5.19703992661925957481520918925, −4.64950704676480919580323590583, −4.17903409931898013180558604652, −3.99633771144638777938101362794, −3.27545247173021532503507514505, −3.25218831129758247552551985199, −2.80966264078545540705206855109, −2.47777112853799117885369613054, −1.85854846336723654086047647703, −1.33396645744727983997076155033, 0, 0, 1.33396645744727983997076155033, 1.85854846336723654086047647703, 2.47777112853799117885369613054, 2.80966264078545540705206855109, 3.25218831129758247552551985199, 3.27545247173021532503507514505, 3.99633771144638777938101362794, 4.17903409931898013180558604652, 4.64950704676480919580323590583, 5.19703992661925957481520918925, 5.52344065907571606240257717444, 5.54381359289116914041628820766, 6.15620996062020129395201056662, 6.56558165991719062802854989718, 7.08270744356990329242129282345, 7.08648667788623873489791526150, 7.75274728439861315120651588333, 7.81906273187618179676538576740