# Properties

 Label 2-45e2-5.4-c1-0-60 Degree $2$ Conductor $2025$ Sign $-0.894 + 0.447i$ Analytic cond. $16.1697$ Root an. cond. $4.02115$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.473i·2-s + 1.77·4-s − 2.56i·7-s − 1.78i·8-s − 6.16·11-s + 2.13i·13-s − 1.21·14-s + 2.70·16-s − 3.16i·17-s − 0.356·19-s + 2.91i·22-s − 4.21i·23-s + 1.00·26-s − 4.55i·28-s − 1.68·29-s + ⋯
 L(s)  = 1 − 0.334i·2-s + 0.888·4-s − 0.968i·7-s − 0.631i·8-s − 1.85·11-s + 0.591i·13-s − 0.324·14-s + 0.676·16-s − 0.768i·17-s − 0.0817·19-s + 0.622i·22-s − 0.878i·23-s + 0.197·26-s − 0.860i·28-s − 0.313·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2025$$    =    $$3^{4} \cdot 5^{2}$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$16.1697$$ Root analytic conductor: $$4.02115$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2025} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2025,\ (\ :1/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.214871502$$ $$L(\frac12)$$ $$\approx$$ $$1.214871502$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 + 0.473iT - 2T^{2}$$
7 $$1 + 2.56iT - 7T^{2}$$
11 $$1 + 6.16T + 11T^{2}$$
13 $$1 - 2.13iT - 13T^{2}$$
17 $$1 + 3.16iT - 17T^{2}$$
19 $$1 + 0.356T + 19T^{2}$$
23 $$1 + 4.21iT - 23T^{2}$$
29 $$1 + 1.68T + 29T^{2}$$
31 $$1 + 8.25T + 31T^{2}$$
37 $$1 + 3.63iT - 37T^{2}$$
41 $$1 + 2.73T + 41T^{2}$$
43 $$1 + 7.67iT - 43T^{2}$$
47 $$1 - 11.4iT - 47T^{2}$$
53 $$1 + 9.43iT - 53T^{2}$$
59 $$1 + 10.2T + 59T^{2}$$
61 $$1 + 0.0109T + 61T^{2}$$
67 $$1 + 0.982iT - 67T^{2}$$
71 $$1 + 6.43T + 71T^{2}$$
73 $$1 - 6.61iT - 73T^{2}$$
79 $$1 - 9.47T + 79T^{2}$$
83 $$1 + 10.4iT - 83T^{2}$$
89 $$1 - 6.26T + 89T^{2}$$
97 $$1 + 7.20iT - 97T^{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

−8.855635671052905757821043207558, −7.67401442976727249469760940112, −7.43135375208477146584867389907, −6.59441024224015423301369040956, −5.60349832486473857510208238489, −4.72509282502336001191454746792, −3.67307300956387166792815452353, −2.74082218441246664874929194333, −1.88614411546273063315691791618, −0.37548616613330096952572053691, 1.79416562186639510609692881839, 2.63679437891508466686820257125, 3.42402840280227718299451747433, 5.03594626113264859829864309036, 5.58361964534460246762532512207, 6.15432827060297137094512554172, 7.29834817684744515787260088608, 7.85568911291664406372467668058, 8.451849935161767410214418666527, 9.422336068137157663075115931790