Properties

 Label 2-2475-11.10-c0-0-1 Degree $2$ Conductor $2475$ Sign $-1$ Analytic cond. $1.23518$ Root an. cond. $1.11138$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 1.41i·2-s − 1.00·4-s + 1.41i·7-s + 11-s + 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·22-s − 2.00·26-s − 1.41i·28-s − 1.41i·32-s + 2.00·34-s + 1.41i·43-s − 1.00·44-s + ⋯
 L(s)  = 1 + 1.41i·2-s − 1.00·4-s + 1.41i·7-s + 11-s + 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·22-s − 2.00·26-s − 1.41i·28-s − 1.41i·32-s + 2.00·34-s + 1.41i·43-s − 1.00·44-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$2475$$    =    $$3^{2} \cdot 5^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$1.23518$$ Root analytic conductor: $$1.11138$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2475} (901, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2475,\ (\ :0),\ -1)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.184399837$$ $$L(\frac12)$$ $$\approx$$ $$1.184399837$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

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

−9.148943117436535000733471307183, −8.793860599828825712313843901958, −7.87279740722943014216247263140, −7.01917485879440151615211308252, −6.47194816248568494855302905191, −5.82854231331736859325744871499, −4.98190229141573149935883669302, −4.31087894259796945565522230331, −2.89133282975621820332282598464, −1.85598143194806943782795038501, 0.830067582061538016029864266451, 1.70428658917981024818608436446, 2.98543465640059473653356308600, 3.86423390662753462415709507789, 4.14989250529904364092095819238, 5.43950039264207885793310037888, 6.50316908836612358804794293084, 7.21442510577589526809801131629, 8.139626582578233471673799587764, 8.897213897461852314347870728101