# Properties

 Label 2-1045-1.1-c1-0-24 Degree $2$ Conductor $1045$ Sign $-1$ Analytic cond. $8.34436$ Root an. cond. $2.88866$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.0881·2-s − 3.20·3-s − 1.99·4-s + 5-s − 0.282·6-s − 1.95·7-s − 0.351·8-s + 7.26·9-s + 0.0881·10-s − 11-s + 6.38·12-s + 3.20·13-s − 0.172·14-s − 3.20·15-s + 3.95·16-s + 0.503·17-s + 0.640·18-s + 19-s − 1.99·20-s + 6.25·21-s − 0.0881·22-s + 1.05·23-s + 1.12·24-s + 25-s + 0.282·26-s − 13.6·27-s + 3.88·28-s + ⋯
 L(s)  = 1 + 0.0623·2-s − 1.84·3-s − 0.996·4-s + 0.447·5-s − 0.115·6-s − 0.737·7-s − 0.124·8-s + 2.42·9-s + 0.0278·10-s − 0.301·11-s + 1.84·12-s + 0.889·13-s − 0.0459·14-s − 0.827·15-s + 0.988·16-s + 0.122·17-s + 0.150·18-s + 0.229·19-s − 0.445·20-s + 1.36·21-s − 0.0187·22-s + 0.220·23-s + 0.230·24-s + 0.200·25-s + 0.0554·26-s − 2.62·27-s + 0.734·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $-1$ Analytic conductor: $$8.34436$$ Root analytic conductor: $$2.88866$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1045,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad5 $$1 - T$$
11 $$1 + T$$
19 $$1 - T$$
good2 $$1 - 0.0881T + 2T^{2}$$
3 $$1 + 3.20T + 3T^{2}$$
7 $$1 + 1.95T + 7T^{2}$$
13 $$1 - 3.20T + 13T^{2}$$
17 $$1 - 0.503T + 17T^{2}$$
23 $$1 - 1.05T + 23T^{2}$$
29 $$1 + 7.91T + 29T^{2}$$
31 $$1 - 9.52T + 31T^{2}$$
37 $$1 + 5.22T + 37T^{2}$$
41 $$1 - 8.32T + 41T^{2}$$
43 $$1 + 8.04T + 43T^{2}$$
47 $$1 + 9.39T + 47T^{2}$$
53 $$1 - 2.37T + 53T^{2}$$
59 $$1 + 2.80T + 59T^{2}$$
61 $$1 + 1.53T + 61T^{2}$$
67 $$1 + 9.79T + 67T^{2}$$
71 $$1 + 4.45T + 71T^{2}$$
73 $$1 + 1.55T + 73T^{2}$$
79 $$1 - 3.00T + 79T^{2}$$
83 $$1 + 2.22T + 83T^{2}$$
89 $$1 - 6.10T + 89T^{2}$$
97 $$1 + 4.69T + 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

−9.808584307325280268850538388518, −8.903310233301945018355952283008, −7.70747774896139184906052044713, −6.57031995808677251812175765834, −5.99488277320995693797376354037, −5.27799974047905390366857456996, −4.50551575743991790104397101702, −3.42197831029460537931350324079, −1.28254953411861682805164698558, 0, 1.28254953411861682805164698558, 3.42197831029460537931350324079, 4.50551575743991790104397101702, 5.27799974047905390366857456996, 5.99488277320995693797376354037, 6.57031995808677251812175765834, 7.70747774896139184906052044713, 8.903310233301945018355952283008, 9.808584307325280268850538388518