# Properties

 Label 2-51842-1.1-c1-0-8 Degree $2$ Conductor $51842$ Sign $-1$ Analytic cond. $413.960$ Root an. cond. $20.3460$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 4·11-s − 2·12-s + 16-s + 6·17-s − 18-s − 6·19-s + 4·22-s + 2·24-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s − 32-s + 8·33-s − 6·34-s + 36-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s − 4·44-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s + 0.408·24-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$51842$$    =    $$2 \cdot 7^{2} \cdot 23^{2}$$ Sign: $-1$ Analytic conductor: $$413.960$$ Root analytic conductor: $$20.3460$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{51842} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 51842,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1 + T$$
7 $$1$$
23 $$1$$
good3 $$1 + 2 T + p T^{2}$$
5 $$1 + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
29 $$1 - 10 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 2 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 + 14 T + p T^{2}$$
97 $$1 + 2 T + p T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−14.81571557318654, −14.31335728427563, −13.71849204895443, −12.92260136984502, −12.53031177781911, −12.20332361592663, −11.46765779014662, −11.13703251478352, −10.57525761394699, −10.13629072447967, −9.843436736458840, −9.005757289350659, −8.309523627842303, −8.011975447433408, −7.411380901946456, −6.753478270101633, −6.131942894743525, −5.761058112041112, −5.226504209956532, −4.583673345243754, −3.869926415833682, −2.898059288679726, −2.472996052957225, −1.498267036194116, −0.7059137775649130, 0, 0.7059137775649130, 1.498267036194116, 2.472996052957225, 2.898059288679726, 3.869926415833682, 4.583673345243754, 5.226504209956532, 5.761058112041112, 6.131942894743525, 6.753478270101633, 7.411380901946456, 8.011975447433408, 8.309523627842303, 9.005757289350659, 9.843436736458840, 10.13629072447967, 10.57525761394699, 11.13703251478352, 11.46765779014662, 12.20332361592663, 12.53031177781911, 12.92260136984502, 13.71849204895443, 14.31335728427563, 14.81571557318654