# Properties

 Label 2-369600-1.1-c1-0-82 Degree $2$ Conductor $369600$ Sign $1$ Analytic cond. $2951.27$ Root an. cond. $54.3256$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s − 7-s + 9-s − 11-s − 6·13-s + 7·17-s + 5·19-s − 21-s + 23-s + 27-s + 5·29-s − 8·31-s − 33-s − 2·37-s − 6·39-s + 12·41-s − 11·43-s − 8·47-s + 49-s + 7·51-s − 11·53-s + 5·57-s + 5·59-s − 7·61-s − 63-s − 2·67-s + 69-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.928·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.960·39-s + 1.87·41-s − 1.67·43-s − 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.51·53-s + 0.662·57-s + 0.650·59-s − 0.896·61-s − 0.125·63-s − 0.244·67-s + 0.120·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$369600$$    =    $$2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Sign: $1$ Analytic conductor: $$2951.27$$ Root analytic conductor: $$54.3256$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{369600} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 369600,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.097711550$$ $$L(\frac12)$$ $$\approx$$ $$2.097711550$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
5 $$1$$
7 $$1 + T$$
11 $$1 + T$$
good13 $$1 + 6 T + p T^{2}$$
17 $$1 - 7 T + p T^{2}$$
19 $$1 - 5 T + p T^{2}$$
23 $$1 - T + p T^{2}$$
29 $$1 - 5 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 12 T + p T^{2}$$
43 $$1 + 11 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 11 T + p T^{2}$$
59 $$1 - 5 T + p T^{2}$$
61 $$1 + 7 T + p T^{2}$$
67 $$1 + 2 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 + 4 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 + T + p T^{2}$$
89 $$1 - 15 T + p T^{2}$$
97 $$1 + 3 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

−12.55288867390874, −12.07860429672488, −11.80714477391643, −11.10470098438029, −10.61961708163846, −10.04601073738302, −9.661805664610268, −9.563929283247534, −8.984290895510486, −8.217667960525244, −7.938481376332968, −7.427284726198312, −7.189445591534587, −6.599957660660689, −5.933422471471162, −5.408911600112916, −4.988721527700073, −4.629335391598211, −3.782414466246946, −3.291949034928340, −2.973098603114373, −2.455153594297835, −1.719797083812183, −1.175767332979478, −0.3647561863329844, 0.3647561863329844, 1.175767332979478, 1.719797083812183, 2.455153594297835, 2.973098603114373, 3.291949034928340, 3.782414466246946, 4.629335391598211, 4.988721527700073, 5.408911600112916, 5.933422471471162, 6.599957660660689, 7.189445591534587, 7.427284726198312, 7.938481376332968, 8.217667960525244, 8.984290895510486, 9.563929283247534, 9.661805664610268, 10.04601073738302, 10.61961708163846, 11.10470098438029, 11.80714477391643, 12.07860429672488, 12.55288867390874