# Properties

 Degree 2 Conductor 619 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.61·2-s + 0.252·3-s + 4.85·4-s + 3.30·5-s − 0.660·6-s − 1.70·7-s − 7.47·8-s − 2.93·9-s − 8.65·10-s + 5.06·11-s + 1.22·12-s + 5.40·13-s + 4.47·14-s + 0.833·15-s + 9.86·16-s − 0.784·17-s + 7.68·18-s + 1.17·19-s + 16.0·20-s − 0.430·21-s − 13.2·22-s − 1.79·23-s − 1.88·24-s + 5.92·25-s − 14.1·26-s − 1.49·27-s − 8.30·28-s + ⋯
 L(s)  = 1 − 1.85·2-s + 0.145·3-s + 2.42·4-s + 1.47·5-s − 0.269·6-s − 0.646·7-s − 2.64·8-s − 0.978·9-s − 2.73·10-s + 1.52·11-s + 0.353·12-s + 1.49·13-s + 1.19·14-s + 0.215·15-s + 2.46·16-s − 0.190·17-s + 1.81·18-s + 0.269·19-s + 3.58·20-s − 0.0940·21-s − 2.82·22-s − 0.374·23-s − 0.384·24-s + 1.18·25-s − 2.77·26-s − 0.288·27-s − 1.56·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$619$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{619} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 619,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.878057$ $L(\frac12)$ $\approx$ $0.878057$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 619$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 619$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad619 $$1 - T$$
good2 $$1 + 2.61T + 2T^{2}$$
3 $$1 - 0.252T + 3T^{2}$$
5 $$1 - 3.30T + 5T^{2}$$
7 $$1 + 1.70T + 7T^{2}$$
11 $$1 - 5.06T + 11T^{2}$$
13 $$1 - 5.40T + 13T^{2}$$
17 $$1 + 0.784T + 17T^{2}$$
19 $$1 - 1.17T + 19T^{2}$$
23 $$1 + 1.79T + 23T^{2}$$
29 $$1 - 7.10T + 29T^{2}$$
31 $$1 + 9.66T + 31T^{2}$$
37 $$1 + 3.34T + 37T^{2}$$
41 $$1 - 8.63T + 41T^{2}$$
43 $$1 - 1.57T + 43T^{2}$$
47 $$1 - 4.16T + 47T^{2}$$
53 $$1 + 1.42T + 53T^{2}$$
59 $$1 - 9.64T + 59T^{2}$$
61 $$1 - 13.8T + 61T^{2}$$
67 $$1 + 10.7T + 67T^{2}$$
71 $$1 + 0.285T + 71T^{2}$$
73 $$1 + 16.4T + 73T^{2}$$
79 $$1 - 7.56T + 79T^{2}$$
83 $$1 + 2.48T + 83T^{2}$$
89 $$1 - 10.8T + 89T^{2}$$
97 $$1 + 0.0677T + 97T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−10.33252087163809259982240458122, −9.511147867875494730166005147942, −8.966901905977217627194341335098, −8.532378731569592699421309085384, −7.11122608790616271868252040268, −6.19945568928341931744043206651, −5.91320127558740713138822347535, −3.46746296393431818013397926608, −2.22007345682876857145305559323, −1.14079133280932028941658238897, 1.14079133280932028941658238897, 2.22007345682876857145305559323, 3.46746296393431818013397926608, 5.91320127558740713138822347535, 6.19945568928341931744043206651, 7.11122608790616271868252040268, 8.532378731569592699421309085384, 8.966901905977217627194341335098, 9.511147867875494730166005147942, 10.33252087163809259982240458122