# Properties

 Degree 2 Conductor $2^{2} \cdot 17 \cdot 59$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.864·3-s − 1.40·5-s − 2.45·7-s − 2.25·9-s + 1.26·11-s + 3.79·13-s − 1.21·15-s − 17-s + 6.52·19-s − 2.12·21-s + 1.32·23-s − 3.02·25-s − 4.54·27-s − 7.10·29-s + 6.91·31-s + 1.09·33-s + 3.45·35-s + 6.38·37-s + 3.27·39-s − 7.21·41-s + 9.21·43-s + 3.16·45-s − 11.3·47-s − 0.958·49-s − 0.864·51-s − 12.1·53-s − 1.78·55-s + ⋯
 L(s)  = 1 + 0.499·3-s − 0.628·5-s − 0.928·7-s − 0.750·9-s + 0.382·11-s + 1.05·13-s − 0.313·15-s − 0.242·17-s + 1.49·19-s − 0.463·21-s + 0.275·23-s − 0.605·25-s − 0.874·27-s − 1.31·29-s + 1.24·31-s + 0.191·33-s + 0.583·35-s + 1.04·37-s + 0.525·39-s − 1.12·41-s + 1.40·43-s + 0.471·45-s − 1.65·47-s − 0.136·49-s − 0.121·51-s − 1.67·53-s − 0.240·55-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4012$$    =    $$2^{2} \cdot 17 \cdot 59$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{4012} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 4012,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;17,\;59\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
17 $$1 + T$$
59 $$1 - T$$
good3 $$1 - 0.864T + 3T^{2}$$
5 $$1 + 1.40T + 5T^{2}$$
7 $$1 + 2.45T + 7T^{2}$$
11 $$1 - 1.26T + 11T^{2}$$
13 $$1 - 3.79T + 13T^{2}$$
19 $$1 - 6.52T + 19T^{2}$$
23 $$1 - 1.32T + 23T^{2}$$
29 $$1 + 7.10T + 29T^{2}$$
31 $$1 - 6.91T + 31T^{2}$$
37 $$1 - 6.38T + 37T^{2}$$
41 $$1 + 7.21T + 41T^{2}$$
43 $$1 - 9.21T + 43T^{2}$$
47 $$1 + 11.3T + 47T^{2}$$
53 $$1 + 12.1T + 53T^{2}$$
61 $$1 + 2.31T + 61T^{2}$$
67 $$1 + 14.7T + 67T^{2}$$
71 $$1 + 2.56T + 71T^{2}$$
73 $$1 - 4.85T + 73T^{2}$$
79 $$1 - 9.22T + 79T^{2}$$
83 $$1 + 12.6T + 83T^{2}$$
89 $$1 - 10.0T + 89T^{2}$$
97 $$1 + 6.67T + 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

−7.961394863218107692502679611125, −7.61674559906131211583254741653, −6.48014215167741697078892743149, −6.05793613825908245247721255540, −5.08556486671857651105882409824, −3.96302100652744609284563407337, −3.38358915436065618449281937130, −2.78469006648003287618148161453, −1.37948633664658586709166330200, 0, 1.37948633664658586709166330200, 2.78469006648003287618148161453, 3.38358915436065618449281937130, 3.96302100652744609284563407337, 5.08556486671857651105882409824, 6.05793613825908245247721255540, 6.48014215167741697078892743149, 7.61674559906131211583254741653, 7.961394863218107692502679611125