# Properties

 Label 2-1-1.1-c17-0-0 Degree $2$ Conductor $1$ Sign $-1$ Analytic cond. $1.83222$ Root an. cond. $1.35359$ Motivic weight $17$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 528·2-s − 4.28e3·3-s + 1.47e5·4-s − 1.02e6·5-s + 2.26e6·6-s + 3.22e6·7-s − 8.78e6·8-s − 1.10e8·9-s + 5.41e8·10-s − 7.53e8·11-s − 6.32e8·12-s + 2.54e9·13-s − 1.70e9·14-s + 4.39e9·15-s − 1.47e10·16-s − 5.42e9·17-s + 5.84e10·18-s + 1.48e9·19-s − 1.51e11·20-s − 1.38e10·21-s + 3.97e11·22-s − 3.17e11·23-s + 3.76e10·24-s + 2.89e11·25-s − 1.34e12·26-s + 1.02e12·27-s + 4.76e11·28-s + ⋯
 L(s)  = 1 − 1.45·2-s − 0.376·3-s + 1.12·4-s − 1.17·5-s + 0.549·6-s + 0.211·7-s − 0.185·8-s − 0.857·9-s + 1.71·10-s − 1.06·11-s − 0.424·12-s + 0.863·13-s − 0.308·14-s + 0.442·15-s − 0.856·16-s − 0.188·17-s + 1.25·18-s + 0.0200·19-s − 1.32·20-s − 0.0797·21-s + 1.54·22-s − 0.844·23-s + 0.0697·24-s + 0.379·25-s − 1.26·26-s + 0.700·27-s + 0.238·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(18-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+17/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1$$ Sign: $-1$ Analytic conductor: $$1.83222$$ Root analytic conductor: $$1.35359$$ Motivic weight: $$17$$ Rational: yes Arithmetic: yes Character: $\chi_{1} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1,\ (\ :17/2),\ -1)$$

## Particular Values

 $$L(9)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{19}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 + 33 p^{4} T + p^{17} T^{2}$$
3 $$1 + 476 p^{2} T + p^{17} T^{2}$$
5 $$1 + 41034 p^{2} T + p^{17} T^{2}$$
7 $$1 - 460856 p T + p^{17} T^{2}$$
11 $$1 + 68510748 p T + p^{17} T^{2}$$
13 $$1 - 195466502 p T + p^{17} T^{2}$$
17 $$1 + 5429742318 T + p^{17} T^{2}$$
19 $$1 - 1487499860 T + p^{17} T^{2}$$
23 $$1 + 317091823464 T + p^{17} T^{2}$$
29 $$1 - 2433410602590 T + p^{17} T^{2}$$
31 $$1 + 8849722053088 T + p^{17} T^{2}$$
37 $$1 - 12691652946662 T + p^{17} T^{2}$$
41 $$1 - 48864151002282 T + p^{17} T^{2}$$
43 $$1 + 91019974317844 T + p^{17} T^{2}$$
47 $$1 + 49304994276048 T + p^{17} T^{2}$$
53 $$1 - 22940453195766 T + p^{17} T^{2}$$
59 $$1 - 32695090729980 T + p^{17} T^{2}$$
61 $$1 + 1308285854869378 T + p^{17} T^{2}$$
67 $$1 - 5196143861984132 T + p^{17} T^{2}$$
71 $$1 + 3709489877412408 T + p^{17} T^{2}$$
73 $$1 - 3402372968272586 T + p^{17} T^{2}$$
79 $$1 - 2366533941308240 T + p^{17} T^{2}$$
83 $$1 + 29766750443172204 T + p^{17} T^{2}$$
89 $$1 - 29167184100574170 T + p^{17} T^{2}$$
97 $$1 + 63769879140957598 T + p^{17} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$