# Properties

 Label 2-230-1.1-c1-0-3 Degree $2$ Conductor $230$ Sign $1$ Analytic cond. $1.83655$ Root an. cond. $1.35519$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 1.79·3-s + 4-s − 5-s − 1.79·6-s + 2.79·7-s − 8-s + 0.208·9-s + 10-s + 3.79·11-s + 1.79·12-s + 1.20·13-s − 2.79·14-s − 1.79·15-s + 16-s − 3.79·17-s − 0.208·18-s + 1.20·19-s − 20-s + 5·21-s − 3.79·22-s + 23-s − 1.79·24-s + 25-s − 1.20·26-s − 5.00·27-s + 2.79·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.03·3-s + 0.5·4-s − 0.447·5-s − 0.731·6-s + 1.05·7-s − 0.353·8-s + 0.0695·9-s + 0.316·10-s + 1.14·11-s + 0.517·12-s + 0.335·13-s − 0.746·14-s − 0.462·15-s + 0.250·16-s − 0.919·17-s − 0.0491·18-s + 0.277·19-s − 0.223·20-s + 1.09·21-s − 0.808·22-s + 0.208·23-s − 0.365·24-s + 0.200·25-s − 0.237·26-s − 0.962·27-s + 0.527·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$230$$    =    $$2 \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$1.83655$$ Root analytic conductor: $$1.35519$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{230} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 230,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.272117732$$ $$L(\frac12)$$ $$\approx$$ $$1.272117732$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 + T$$
23 $$1 - T$$
good3 $$1 - 1.79T + 3T^{2}$$
7 $$1 - 2.79T + 7T^{2}$$
11 $$1 - 3.79T + 11T^{2}$$
13 $$1 - 1.20T + 13T^{2}$$
17 $$1 + 3.79T + 17T^{2}$$
19 $$1 - 1.20T + 19T^{2}$$
29 $$1 + 1.58T + 29T^{2}$$
31 $$1 - 10.3T + 31T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 + 2.20T + 41T^{2}$$
43 $$1 + 7.16T + 43T^{2}$$
47 $$1 + 13.5T + 47T^{2}$$
53 $$1 - 6T + 53T^{2}$$
59 $$1 + 4.41T + 59T^{2}$$
61 $$1 + 3.37T + 61T^{2}$$
67 $$1 + 7.16T + 67T^{2}$$
71 $$1 + 5.37T + 71T^{2}$$
73 $$1 + 14.7T + 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 + 3.16T + 89T^{2}$$
97 $$1 - 14.9T + 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

−11.74263410731868075600580313835, −11.41971376748791719919366959666, −10.10507068480568508870022571273, −8.891536565685023601694878685968, −8.501411878293945551716222971506, −7.57143152749856172316050467033, −6.41055578761420862138865627363, −4.60961578868387518096880916533, −3.25470319294818186689101430882, −1.69968985125155088025702730602, 1.69968985125155088025702730602, 3.25470319294818186689101430882, 4.60961578868387518096880916533, 6.41055578761420862138865627363, 7.57143152749856172316050467033, 8.501411878293945551716222971506, 8.891536565685023601694878685968, 10.10507068480568508870022571273, 11.41971376748791719919366959666, 11.74263410731868075600580313835