# Properties

 Label 2-42e2-1.1-c3-0-41 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 + 89·13-s − 163·19-s − 125·25-s − 19·31-s − 433·37-s + 449·43-s + 182·61-s + 1.00e3·67-s − 919·73-s + 503·79-s − 1.33e3·97-s − 19·103-s − 1.56e3·109-s + ⋯
 L(s)  = 1 + 1.89·13-s − 1.96·19-s − 25-s − 0.110·31-s − 1.92·37-s + 1.59·43-s + 0.382·61-s + 1.83·67-s − 1.47·73-s + 0.716·79-s − 1.39·97-s − 0.0181·103-s − 1.37·109-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 + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 - 89 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + 163 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 + 19 T + p^{3} T^{2}$$
37 $$1 + 433 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 - 449 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 - 182 T + p^{3} T^{2}$$
67 $$1 - 1007 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 + 919 T + p^{3} T^{2}$$
79 $$1 - 503 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + 1330 T + p^{3} 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

−8.584050236184648241912289015518, −7.923691641783472199781348495558, −6.78672909301721518149033887856, −6.18720721635831425964345185466, −5.40816605755441490751015948002, −4.14984695869246579901456545077, −3.67019387035805425678120614093, −2.32800845535865852702276396112, −1.34516728263619274132007212632, 0, 1.34516728263619274132007212632, 2.32800845535865852702276396112, 3.67019387035805425678120614093, 4.14984695869246579901456545077, 5.40816605755441490751015948002, 6.18720721635831425964345185466, 6.78672909301721518149033887856, 7.923691641783472199781348495558, 8.584050236184648241912289015518