# Properties

 Label 4-353-1.1-c1e2-0-0 Degree $4$ Conductor $353$ Sign $1$ Analytic cond. $0.0225075$ Root an. cond. $0.387330$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s − 2·4-s + 5-s + 2·6-s + 3·8-s − 10-s + 2·11-s + 4·12-s − 13-s − 2·15-s + 16-s + 6·17-s − 6·19-s − 2·20-s − 2·22-s − 2·23-s − 6·24-s − 25-s + 26-s + 2·27-s − 2·29-s + 2·30-s − 4·31-s − 2·32-s − 4·33-s − 6·34-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s − 4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s − 0.316·10-s + 0.603·11-s + 1.15·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.447·20-s − 0.426·22-s − 0.417·23-s − 1.22·24-s − 1/5·25-s + 0.196·26-s + 0.384·27-s − 0.371·29-s + 0.365·30-s − 0.718·31-s − 0.353·32-s − 0.696·33-s − 1.02·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$353$$ Sign: $1$ Analytic conductor: $$0.0225075$$ Root analytic conductor: $$0.387330$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 353,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1859131786$$ $$L(\frac12)$$ $$\approx$$ $$0.1859131786$$ $$L(\frac{3}{2})$$ 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)$
bad353$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 9 T + p T^{2} )$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}$$
7$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
11$D_{4}$ $$1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} )$$
19$D_{4}$ $$1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$D_{4}$ $$1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 23 T^{2} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
89$D_{4}$ $$1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 11 T + 116 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
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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

−19.9961442661, −19.3214919249, −18.7976671865, −18.2092074740, −17.6758825494, −17.1141440050, −16.8700603189, −16.3062469438, −15.0721190988, −14.3933411178, −13.8616163752, −12.9097577259, −12.3526386620, −11.5059400610, −10.7395302821, −9.94383563180, −9.33983492188, −8.54105304961, −7.57074114369, −6.16753924570, −5.48946407918, −4.23852001252, 4.23852001252, 5.48946407918, 6.16753924570, 7.57074114369, 8.54105304961, 9.33983492188, 9.94383563180, 10.7395302821, 11.5059400610, 12.3526386620, 12.9097577259, 13.8616163752, 14.3933411178, 15.0721190988, 16.3062469438, 16.8700603189, 17.1141440050, 17.6758825494, 18.2092074740, 18.7976671865, 19.3214919249, 19.9961442661