# Properties

 Label 2-1-1.1-c15-0-0 Degree $2$ Conductor $1$ Sign $1$ Analytic cond. $1.42693$ Root an. cond. $1.19454$ Motivic weight $15$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 216·2-s − 3.34e3·3-s + 1.38e4·4-s + 5.21e4·5-s − 7.23e5·6-s + 2.82e6·7-s − 4.07e6·8-s − 3.13e6·9-s + 1.12e7·10-s + 2.05e7·11-s − 4.64e7·12-s − 1.90e8·13-s + 6.09e8·14-s − 1.74e8·15-s − 1.33e9·16-s + 1.64e9·17-s − 6.78e8·18-s + 1.56e9·19-s + 7.23e8·20-s − 9.44e9·21-s + 4.44e9·22-s + 9.45e9·23-s + 1.36e10·24-s − 2.78e10·25-s − 4.10e10·26-s + 5.85e10·27-s + 3.91e10·28-s + ⋯
 L(s)  = 1 + 1.19·2-s − 0.883·3-s + 0.423·4-s + 0.298·5-s − 1.05·6-s + 1.29·7-s − 0.687·8-s − 0.218·9-s + 0.355·10-s + 0.318·11-s − 0.374·12-s − 0.840·13-s + 1.54·14-s − 0.263·15-s − 1.24·16-s + 0.973·17-s − 0.261·18-s + 0.401·19-s + 0.126·20-s − 1.14·21-s + 0.380·22-s + 0.578·23-s + 0.607·24-s − 0.911·25-s − 1.00·26-s + 1.07·27-s + 0.549·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+15/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1$$ Sign: $1$ Analytic conductor: $$1.42693$$ Root analytic conductor: $$1.19454$$ Motivic weight: $$15$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1,\ (\ :15/2),\ 1)$$

## Particular Values

 $$L(8)$$ $$\approx$$ $$1.520561669$$ $$L(\frac12)$$ $$\approx$$ $$1.520561669$$ $$L(\frac{17}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 - 27 p^{3} T + p^{15} T^{2}$$
3 $$1 + 124 p^{3} T + p^{15} T^{2}$$
5 $$1 - 10422 p T + p^{15} T^{2}$$
7 $$1 - 403208 p T + p^{15} T^{2}$$
11 $$1 - 1871532 p T + p^{15} T^{2}$$
13 $$1 + 14621026 p T + p^{15} T^{2}$$
17 $$1 - 1646527986 T + p^{15} T^{2}$$
19 $$1 - 1563257180 T + p^{15} T^{2}$$
23 $$1 - 9451116072 T + p^{15} T^{2}$$
29 $$1 + 36902568330 T + p^{15} T^{2}$$
31 $$1 - 71588483552 T + p^{15} T^{2}$$
37 $$1 + 1033652081554 T + p^{15} T^{2}$$
41 $$1 - 1641974018202 T + p^{15} T^{2}$$
43 $$1 + 492403109308 T + p^{15} T^{2}$$
47 $$1 + 3410684952624 T + p^{15} T^{2}$$
53 $$1 - 6797151655902 T + p^{15} T^{2}$$
59 $$1 - 167099268060 p T + p^{15} T^{2}$$
61 $$1 - 4931842626902 T + p^{15} T^{2}$$
67 $$1 + 28837826625364 T + p^{15} T^{2}$$
71 $$1 - 125050114914552 T + p^{15} T^{2}$$
73 $$1 + 82171455513478 T + p^{15} T^{2}$$
79 $$1 + 25413078694480 T + p^{15} T^{2}$$
83 $$1 + 281736730890468 T + p^{15} T^{2}$$
89 $$1 - 715618564776810 T + p^{15} T^{2}$$
97 $$1 - 612786136081826 T + p^{15} 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

−31.31083306980598975354369589119, −29.69484151629079972214366719026, −27.56748408646213265713341304974, −24.35638841200895818548143833510, −22.83359228719678482192637867477, −21.26112311281979572877543178918, −17.51673159287656967480964579195, −14.40110767072392844184488566452, −11.82395546014304725015828959152, −5.26502022758204893749954155127, 5.26502022758204893749954155127, 11.82395546014304725015828959152, 14.40110767072392844184488566452, 17.51673159287656967480964579195, 21.26112311281979572877543178918, 22.83359228719678482192637867477, 24.35638841200895818548143833510, 27.56748408646213265713341304974, 29.69484151629079972214366719026, 31.31083306980598975354369589119