# Properties

 Label 2-42e2-1.1-c3-0-44 Degree $2$ Conductor $1764$ Sign $-1$ Analytic cond. $104.079$ Root an. cond. $10.2019$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 6·5-s + 12·11-s + 82·13-s − 30·17-s − 68·19-s − 216·23-s − 89·25-s − 246·29-s + 112·31-s + 110·37-s − 246·41-s − 172·43-s + 192·47-s − 558·53-s + 72·55-s + 540·59-s − 110·61-s + 492·65-s + 140·67-s + 840·71-s + 550·73-s − 208·79-s + 516·83-s − 180·85-s − 1.39e3·89-s − 408·95-s − 1.58e3·97-s + ⋯
 L(s)  = 1 + 0.536·5-s + 0.328·11-s + 1.74·13-s − 0.428·17-s − 0.821·19-s − 1.95·23-s − 0.711·25-s − 1.57·29-s + 0.648·31-s + 0.488·37-s − 0.937·41-s − 0.609·43-s + 0.595·47-s − 1.44·53-s + 0.176·55-s + 1.19·59-s − 0.230·61-s + 0.938·65-s + 0.255·67-s + 1.40·71-s + 0.881·73-s − 0.296·79-s + 0.682·83-s − 0.229·85-s − 1.66·89-s − 0.440·95-s − 1.66·97-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$104.079$$ Root analytic conductor: $$10.2019$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{1764} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1764,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1$$
good5 $$1 - 6 T + p^{3} T^{2}$$
11 $$1 - 12 T + p^{3} T^{2}$$
13 $$1 - 82 T + p^{3} T^{2}$$
17 $$1 + 30 T + p^{3} T^{2}$$
19 $$1 + 68 T + p^{3} T^{2}$$
23 $$1 + 216 T + p^{3} T^{2}$$
29 $$1 + 246 T + p^{3} T^{2}$$
31 $$1 - 112 T + p^{3} T^{2}$$
37 $$1 - 110 T + p^{3} T^{2}$$
41 $$1 + 6 p T + p^{3} T^{2}$$
43 $$1 + 4 p T + p^{3} T^{2}$$
47 $$1 - 192 T + p^{3} T^{2}$$
53 $$1 + 558 T + p^{3} T^{2}$$
59 $$1 - 540 T + p^{3} T^{2}$$
61 $$1 + 110 T + p^{3} T^{2}$$
67 $$1 - 140 T + p^{3} T^{2}$$
71 $$1 - 840 T + p^{3} T^{2}$$
73 $$1 - 550 T + p^{3} T^{2}$$
79 $$1 + 208 T + p^{3} T^{2}$$
83 $$1 - 516 T + p^{3} T^{2}$$
89 $$1 + 1398 T + p^{3} T^{2}$$
97 $$1 + 1586 T + p^{3} T^{2}$$
show less
$$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

−8.438762494670362863964694086153, −8.007065024776906709255659445450, −6.69983176620356646016116611703, −6.15722172992460253299256757843, −5.49826090771971409946405709650, −4.16189100294200809516396173417, −3.65962886513834044890905794259, −2.21734621807336752729527798061, −1.46518633748361488360563753889, 0, 1.46518633748361488360563753889, 2.21734621807336752729527798061, 3.65962886513834044890905794259, 4.16189100294200809516396173417, 5.49826090771971409946405709650, 6.15722172992460253299256757843, 6.69983176620356646016116611703, 8.007065024776906709255659445450, 8.438762494670362863964694086153