# Properties

 Label 2-1840-1.1-c1-0-6 Degree $2$ Conductor $1840$ Sign $1$ Analytic cond. $14.6924$ Root an. cond. $3.83307$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 0.568·3-s − 5-s − 4.73·7-s − 2.67·9-s + 0.360·11-s + 5.26·13-s − 0.568·15-s + 0.370·17-s + 4.60·19-s − 2.69·21-s + 23-s + 25-s − 3.22·27-s + 0.939·29-s − 9.66·31-s + 0.204·33-s + 4.73·35-s + 3.26·37-s + 2.99·39-s + 5.29·41-s + 2.67·45-s + 1.25·47-s + 15.4·49-s + 0.210·51-s + 10.9·53-s − 0.360·55-s + 2.61·57-s + ⋯
 L(s)  = 1 + 0.328·3-s − 0.447·5-s − 1.79·7-s − 0.892·9-s + 0.108·11-s + 1.45·13-s − 0.146·15-s + 0.0899·17-s + 1.05·19-s − 0.587·21-s + 0.208·23-s + 0.200·25-s − 0.620·27-s + 0.174·29-s − 1.73·31-s + 0.0356·33-s + 0.800·35-s + 0.537·37-s + 0.478·39-s + 0.827·41-s + 0.399·45-s + 0.183·47-s + 2.20·49-s + 0.0295·51-s + 1.50·53-s − 0.0486·55-s + 0.346·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1840$$    =    $$2^{4} \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$14.6924$$ Root analytic conductor: $$3.83307$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1840} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1840,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.281066291$$ $$L(\frac12)$$ $$\approx$$ $$1.281066291$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + T$$
23 $$1 - T$$
good3 $$1 - 0.568T + 3T^{2}$$
7 $$1 + 4.73T + 7T^{2}$$
11 $$1 - 0.360T + 11T^{2}$$
13 $$1 - 5.26T + 13T^{2}$$
17 $$1 - 0.370T + 17T^{2}$$
19 $$1 - 4.60T + 19T^{2}$$
29 $$1 - 0.939T + 29T^{2}$$
31 $$1 + 9.66T + 31T^{2}$$
37 $$1 - 3.26T + 37T^{2}$$
41 $$1 - 5.29T + 41T^{2}$$
43 $$1 + 43T^{2}$$
47 $$1 - 1.25T + 47T^{2}$$
53 $$1 - 10.9T + 53T^{2}$$
59 $$1 - 9.66T + 59T^{2}$$
61 $$1 - 9.71T + 61T^{2}$$
67 $$1 - 7.07T + 67T^{2}$$
71 $$1 + 11.3T + 71T^{2}$$
73 $$1 - 0.745T + 73T^{2}$$
79 $$1 + 0.415T + 79T^{2}$$
83 $$1 - 9.26T + 83T^{2}$$
89 $$1 - 12.6T + 89T^{2}$$
97 $$1 - 14.0T + 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

−9.109959738712378489852191146317, −8.693304939406647064969092670324, −7.66031533688569428577097904332, −6.86974555955036521775890010158, −6.03835507846753314531492757952, −5.45116032660235916260362165796, −3.78412579682264420598315908340, −3.50498762785343705309704084704, −2.54526778852463296582397838950, −0.73930114888484845428126629533, 0.73930114888484845428126629533, 2.54526778852463296582397838950, 3.50498762785343705309704084704, 3.78412579682264420598315908340, 5.45116032660235916260362165796, 6.03835507846753314531492757952, 6.86974555955036521775890010158, 7.66031533688569428577097904332, 8.693304939406647064969092670324, 9.109959738712378489852191146317