# Properties

 Label 2-5610-1.1-c1-0-31 Degree $2$ Conductor $5610$ Sign $-1$ Analytic cond. $44.7960$ Root an. cond. $6.69298$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s − 5.12·7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 3.12·13-s + 5.12·14-s + 15-s + 16-s − 17-s − 18-s − 5.12·19-s − 20-s + 5.12·21-s + 22-s + 5.12·23-s + 24-s + 25-s + 3.12·26-s − 27-s − 5.12·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.93·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.866·13-s + 1.36·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.17·19-s − 0.223·20-s + 1.11·21-s + 0.213·22-s + 1.06·23-s + 0.204·24-s + 0.200·25-s + 0.612·26-s − 0.192·27-s − 0.968·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5610$$    =    $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Sign: $-1$ Analytic conductor: $$44.7960$$ Root analytic conductor: $$6.69298$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 5610,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
5 $$1 + T$$
11 $$1 + T$$
17 $$1 + T$$
good7 $$1 + 5.12T + 7T^{2}$$
13 $$1 + 3.12T + 13T^{2}$$
19 $$1 + 5.12T + 19T^{2}$$
23 $$1 - 5.12T + 23T^{2}$$
29 $$1 - 8.24T + 29T^{2}$$
31 $$1 - 6.24T + 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 - 10T + 41T^{2}$$
43 $$1 - 1.12T + 43T^{2}$$
47 $$1 + 4T + 47T^{2}$$
53 $$1 + 0.876T + 53T^{2}$$
59 $$1 - 1.12T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 + 13.1T + 71T^{2}$$
73 $$1 - 7.12T + 73T^{2}$$
79 $$1 - 2.87T + 79T^{2}$$
83 $$1 - 4T + 83T^{2}$$
89 $$1 - 4.24T + 89T^{2}$$
97 $$1 - 8.24T + 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

−7.72579528193371846491705213349, −6.93941952073029132568959007777, −6.52161677746673874930792949851, −5.95847921420948554713859963582, −4.85278165459810449003758763802, −4.09793630490715019314108975147, −2.97532646919772594957301820161, −2.53261331735217934787494582682, −0.849150571873832115931609819761, 0, 0.849150571873832115931609819761, 2.53261331735217934787494582682, 2.97532646919772594957301820161, 4.09793630490715019314108975147, 4.85278165459810449003758763802, 5.95847921420948554713859963582, 6.52161677746673874930792949851, 6.93941952073029132568959007777, 7.72579528193371846491705213349