# Properties

 Degree 2 Conductor $2^{2} \cdot 17 \cdot 59$ Sign $0.994 + 0.108i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Learn more about

## Dirichlet series

 L(s)  = 1 + 0.405i·3-s − 1.20i·5-s + 0.122i·7-s + 2.83·9-s − 1.65i·11-s − 1.21·13-s + 0.488·15-s + (−4.09 − 0.446i)17-s + 5.49·19-s − 0.0496·21-s + 1.29i·23-s + 3.54·25-s + 2.36i·27-s + 9.31i·29-s + 6.06i·31-s + ⋯
 L(s)  = 1 + 0.233i·3-s − 0.538i·5-s + 0.0462i·7-s + 0.945·9-s − 0.498i·11-s − 0.337·13-s + 0.126·15-s + (−0.994 − 0.108i)17-s + 1.26·19-s − 0.0108·21-s + 0.269i·23-s + 0.709·25-s + 0.454i·27-s + 1.72i·29-s + 1.08i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4012$$    =    $$2^{2} \cdot 17 \cdot 59$$ $$\varepsilon$$ = $0.994 + 0.108i$ motivic weight = $$1$$ character : $\chi_{4012} (237, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4012,\ (\ :1/2),\ 0.994 + 0.108i)$ $L(1)$ $\approx$ $2.063342975$ $L(\frac12)$ $\approx$ $2.063342975$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;17,\;59\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + (4.09 + 0.446i)T$$
59 $$1 - T$$
good3 $$1 - 0.405iT - 3T^{2}$$
5 $$1 + 1.20iT - 5T^{2}$$
7 $$1 - 0.122iT - 7T^{2}$$
11 $$1 + 1.65iT - 11T^{2}$$
13 $$1 + 1.21T + 13T^{2}$$
19 $$1 - 5.49T + 19T^{2}$$
23 $$1 - 1.29iT - 23T^{2}$$
29 $$1 - 9.31iT - 29T^{2}$$
31 $$1 - 6.06iT - 31T^{2}$$
37 $$1 + 4.07iT - 37T^{2}$$
41 $$1 + 1.98iT - 41T^{2}$$
43 $$1 - 7.44T + 43T^{2}$$
47 $$1 + 10.1T + 47T^{2}$$
53 $$1 - 4.71T + 53T^{2}$$
61 $$1 + 3.86iT - 61T^{2}$$
67 $$1 - 2.82T + 67T^{2}$$
71 $$1 + 12.5iT - 71T^{2}$$
73 $$1 + 14.7iT - 73T^{2}$$
79 $$1 - 0.917iT - 79T^{2}$$
83 $$1 - 12.7T + 83T^{2}$$
89 $$1 + 2.24T + 89T^{2}$$
97 $$1 - 7.92iT - 97T^{2}$$
show more
show less
\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

−8.659862785577928638021377055377, −7.56544583909486396342494426710, −7.09321705918061736925479074617, −6.27012217514218614859013690239, −5.07698757126377604428298127494, −4.94756991084700169980325135003, −3.81956612161386746582646232450, −3.07650724139413755186994393918, −1.82368785904685883874723491503, −0.841086007343673728656083352250, 0.854785282133091313732816738170, 2.09665753356158987836180234228, 2.79398876239553516663910194757, 4.02161268257019146879878187593, 4.52693977025803491650515308216, 5.53031230478764788026741578549, 6.45540835181264792107260901428, 7.02227648630797534247322642798, 7.58681994783303098253091732845, 8.310527983973165346602173296185