# Properties

 Label 2-39984-1.1-c1-0-88 Degree $2$ Conductor $39984$ Sign $-1$ Analytic cond. $319.273$ Root an. cond. $17.8682$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 3-s + 2·5-s + 9-s + 6·13-s − 2·15-s − 17-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 10·37-s − 6·39-s + 6·41-s − 12·43-s + 2·45-s + 51-s − 10·53-s − 8·59-s − 6·61-s + 12·65-s − 12·67-s − 8·69-s + 6·73-s + 75-s + 8·79-s + 81-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.242·17-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.960·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 0.140·51-s − 1.37·53-s − 1.04·59-s − 0.768·61-s + 1.48·65-s − 1.46·67-s − 0.963·69-s + 0.702·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$39984$$    =    $$2^{4} \cdot 3 \cdot 7^{2} \cdot 17$$ Sign: $-1$ Analytic conductor: $$319.273$$ Root analytic conductor: $$17.8682$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{39984} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 39984,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1 + T$$
7 $$1$$
17 $$1 + T$$
good5 $$1 - 2 T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 + 8 T + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 + 2 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

−15.05321446187649, −14.60930529477696, −13.81078965575082, −13.41992216027841, −12.97065873710832, −12.72076592488550, −11.75245090680044, −11.27700149848067, −10.84346580705203, −10.56606359827229, −9.656736406251172, −9.146506335105158, −9.013524877742196, −7.972356152627851, −7.582701399159548, −6.738631377994659, −6.217039481478766, −5.934280437774619, −5.212748399956355, −4.733054392187898, −3.828771656301505, −3.354992931069669, −2.457674062743786, −1.580237168073311, −1.189180276005915, 0, 1.189180276005915, 1.580237168073311, 2.457674062743786, 3.354992931069669, 3.828771656301505, 4.733054392187898, 5.212748399956355, 5.934280437774619, 6.217039481478766, 6.738631377994659, 7.582701399159548, 7.972356152627851, 9.013524877742196, 9.146506335105158, 9.656736406251172, 10.56606359827229, 10.84346580705203, 11.27700149848067, 11.75245090680044, 12.72076592488550, 12.97065873710832, 13.41992216027841, 13.81078965575082, 14.60930529477696, 15.05321446187649