# Properties

 Degree $2$ Conductor $2151$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯
 L(s)  = 1 − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2151$$    =    $$3^{2} \cdot 239$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{2151} (955, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2151,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8490923334$$ $$L(\frac12)$$ $$\approx$$ $$0.8490923334$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
239 $$1 - T$$
good2 $$1 + T + T^{2}$$
5 $$1 - T + T^{2}$$
7 $$1 + 1.41iT - T^{2}$$
11 $$1 - T + T^{2}$$
13 $$1 - 1.41iT - T^{2}$$
17 $$1 - T + T^{2}$$
19 $$1 - 1.41iT - T^{2}$$
23 $$1 - 1.41iT - T^{2}$$
29 $$1 - T + T^{2}$$
31 $$1 + T + T^{2}$$
37 $$1 + 1.41iT - T^{2}$$
41 $$1 + 1.41iT - T^{2}$$
43 $$1 - T^{2}$$
47 $$1 - 1.41iT - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 + T + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - 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

−9.403900991727383454971185836199, −8.735792119741634860765562326784, −7.58930044922002904632021011796, −7.26809148398465079991793981317, −6.28936765790551897519371521509, −5.42078580743045591351431096127, −4.15390296872720246988521497798, −3.75070727266357459269629251941, −1.78027265371229092511307052832, −1.28936431270954069317179779520, 1.07026718400937328860511981678, 2.25127978319250414000309853719, 3.12570361017273950383304364602, 4.67576576351142655721747271211, 5.38510191980781384452550227640, 6.16191576010009351777978057046, 6.96484129443832487962415621855, 8.134508540993616110825691166980, 8.613284827665564718178877719368, 9.208704971631530789653962510550