# Properties

 Label 2-3450-1.1-c1-0-66 Degree $2$ Conductor $3450$ Sign $-1$ Analytic cond. $27.5483$ Root an. cond. $5.24865$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 2·11-s + 12-s − 4·14-s + 16-s − 2·17-s + 18-s − 4·21-s − 2·22-s − 23-s + 24-s + 27-s − 4·28-s − 4·29-s + 32-s − 2·33-s − 2·34-s + 36-s − 10·37-s + 6·41-s − 4·42-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.872·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.742·29-s + 0.176·32-s − 0.348·33-s − 0.342·34-s + 1/6·36-s − 1.64·37-s + 0.937·41-s − 0.617·42-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3450$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 23$$ Sign: $-1$ Analytic conductor: $$27.5483$$ Root analytic conductor: $$5.24865$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{3450} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 3450,\ (\ :1/2),\ -1)$$

## 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$$
23 $$1 + T$$
good7 $$1 + 4 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
29 $$1 + 4 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 14 T + p T^{2}$$
67 $$1 + 2 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 8 T + p T^{2}$$
89 $$1 + 8 T + p T^{2}$$
97 $$1 + p 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

−8.168420849840029793853910944578, −7.31310319662070416391959014907, −6.69762936587867738080522762207, −6.00729390501890278323839728611, −5.16107792495912297960628169571, −4.19620027639044251554847993934, −3.37748149946480537753655281631, −2.84403763116542600292068150831, −1.81680622419565840347383452699, 0, 1.81680622419565840347383452699, 2.84403763116542600292068150831, 3.37748149946480537753655281631, 4.19620027639044251554847993934, 5.16107792495912297960628169571, 6.00729390501890278323839728611, 6.69762936587867738080522762207, 7.31310319662070416391959014907, 8.168420849840029793853910944578