# Properties

 Label 2-2550-5.4-c1-0-2 Degree $2$ Conductor $2550$ Sign $-0.894 - 0.447i$ Analytic cond. $20.3618$ Root an. cond. $4.51241$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − i·2-s + i·3-s − 4-s + 6-s + 4.89i·7-s + i·8-s − 9-s − i·12-s + 2.89i·13-s + 4.89·14-s + 16-s − i·17-s + i·18-s − 4·19-s − 4.89·21-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.85i·7-s + 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.804i·13-s + 1.30·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s − 1.06·21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{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: $$2550$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 17$$ Sign: $-0.894 - 0.447i$ Analytic conductor: $$20.3618$$ Root analytic conductor: $$4.51241$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2550} (2449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2550,\ (\ :1/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7753025667$$ $$L(\frac12)$$ $$\approx$$ $$0.7753025667$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 - iT$$
5 $$1$$
17 $$1 + iT$$
good7 $$1 - 4.89iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 2.89iT - 13T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 - 6iT - 37T^{2}$$
41 $$1 - 6.89T + 41T^{2}$$
43 $$1 - 0.898iT - 43T^{2}$$
47 $$1 + 9.79iT - 47T^{2}$$
53 $$1 + 11.7iT - 53T^{2}$$
59 $$1 - 4.89T + 59T^{2}$$
61 $$1 + 7.79T + 61T^{2}$$
67 $$1 + 8.89iT - 67T^{2}$$
71 $$1 - 0.898T + 71T^{2}$$
73 $$1 - 1.10iT - 73T^{2}$$
79 $$1 + 13.7T + 79T^{2}$$
83 $$1 - 5.79iT - 83T^{2}$$
89 $$1 + 11.7T + 89T^{2}$$
97 $$1 - 16.6iT - 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

−9.383491671912415463759724948106, −8.655318112539180948485352828815, −8.120910477083480910603916572807, −6.78527493986793413657171787058, −5.87987691284628707246006574945, −5.25227805831730867440728803842, −4.42950169149649833523900349573, −3.46063575178426086665261378584, −2.52881151657490321322747068675, −1.80624680830123940465618460201, 0.26362376804872459902107068556, 1.29040103228238986982684350441, 2.80885593434393858770740299818, 4.01559655696733607028704117303, 4.46726294443081497500882844724, 5.71308281736953227703399701180, 6.36011126016323038697867435359, 7.21552098006507615737752315395, 7.60704981894494113255565636704, 8.288840830487324470528551343664