# Properties

 Label 2-1083-1.1-c3-0-73 Degree $2$ Conductor $1083$ Sign $-1$ Analytic cond. $63.8990$ Root an. cond. $7.99368$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 3·3-s − 7·4-s − 12·5-s − 3·6-s − 20·7-s − 15·8-s + 9·9-s − 12·10-s − 4·11-s + 21·12-s + 76·13-s − 20·14-s + 36·15-s + 41·16-s + 22·17-s + 9·18-s + 84·20-s + 60·21-s − 4·22-s + 82·23-s + 45·24-s + 19·25-s + 76·26-s − 27·27-s + 140·28-s − 242·29-s + ⋯
 L(s)  = 1 + 0.353·2-s − 0.577·3-s − 7/8·4-s − 1.07·5-s − 0.204·6-s − 1.07·7-s − 0.662·8-s + 1/3·9-s − 0.379·10-s − 0.109·11-s + 0.505·12-s + 1.62·13-s − 0.381·14-s + 0.619·15-s + 0.640·16-s + 0.313·17-s + 0.117·18-s + 0.939·20-s + 0.623·21-s − 0.0387·22-s + 0.743·23-s + 0.382·24-s + 0.151·25-s + 0.573·26-s − 0.192·27-s + 0.944·28-s − 1.54·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1083$$    =    $$3 \cdot 19^{2}$$ Sign: $-1$ Analytic conductor: $$63.8990$$ Root analytic conductor: $$7.99368$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1083,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad3 $$1 + p T$$
19 $$1$$
good2 $$1 - T + p^{3} T^{2}$$
5 $$1 + 12 T + p^{3} T^{2}$$
7 $$1 + 20 T + p^{3} T^{2}$$
11 $$1 + 4 T + p^{3} T^{2}$$
13 $$1 - 76 T + p^{3} T^{2}$$
17 $$1 - 22 T + p^{3} T^{2}$$
23 $$1 - 82 T + p^{3} T^{2}$$
29 $$1 + 242 T + p^{3} T^{2}$$
31 $$1 - 126 T + p^{3} T^{2}$$
37 $$1 - 180 T + p^{3} T^{2}$$
41 $$1 - 390 T + p^{3} T^{2}$$
43 $$1 - 308 T + p^{3} T^{2}$$
47 $$1 + 522 T + p^{3} T^{2}$$
53 $$1 - 70 T + p^{3} T^{2}$$
59 $$1 + 188 T + p^{3} T^{2}$$
61 $$1 + 706 T + p^{3} T^{2}$$
67 $$1 + 104 T + p^{3} T^{2}$$
71 $$1 - 432 T + p^{3} T^{2}$$
73 $$1 - 718 T + p^{3} T^{2}$$
79 $$1 + 94 T + p^{3} T^{2}$$
83 $$1 + 1296 T + p^{3} T^{2}$$
89 $$1 + 846 T + p^{3} T^{2}$$
97 $$1 + 830 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

−9.160271589402270976408109520017, −8.249962788334443005741485354870, −7.44081481822011889091134399463, −6.26919706364655544176094888769, −5.75525767772265521607407717880, −4.54961899759875406004252006029, −3.80244219032152589562823617417, −3.16252768986189697423571496283, −0.972523399241738064682902762509, 0, 0.972523399241738064682902762509, 3.16252768986189697423571496283, 3.80244219032152589562823617417, 4.54961899759875406004252006029, 5.75525767772265521607407717880, 6.26919706364655544176094888769, 7.44081481822011889091134399463, 8.249962788334443005741485354870, 9.160271589402270976408109520017