# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 67$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + 5-s − 3.96·7-s + 8-s + 10-s − 3.04·11-s + 0.919·13-s − 3.96·14-s + 16-s − 0.351·17-s − 0.351·19-s + 20-s − 3.04·22-s + 3.43·23-s + 25-s + 0.919·26-s − 3.96·28-s − 9.37·29-s + 2.91·31-s + 32-s − 0.351·34-s − 3.96·35-s + 5.40·37-s − 0.351·38-s + 40-s + 8.45·41-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.50·7-s + 0.353·8-s + 0.316·10-s − 0.919·11-s + 0.254·13-s − 1.06·14-s + 0.250·16-s − 0.0853·17-s − 0.0807·19-s + 0.223·20-s − 0.650·22-s + 0.715·23-s + 0.200·25-s + 0.180·26-s − 0.750·28-s − 1.74·29-s + 0.524·31-s + 0.176·32-s − 0.0603·34-s − 0.670·35-s + 0.888·37-s − 0.0570·38-s + 0.158·40-s + 1.31·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6030$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 67$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{6030} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 6030,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$2.562739145$$ $$L(\frac12)$$ $$\approx$$ $$2.562739145$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;67\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1$$
5 $$1 - T$$
67 $$1 + T$$
good7 $$1 + 3.96T + 7T^{2}$$
11 $$1 + 3.04T + 11T^{2}$$
13 $$1 - 0.919T + 13T^{2}$$
17 $$1 + 0.351T + 17T^{2}$$
19 $$1 + 0.351T + 19T^{2}$$
23 $$1 - 3.43T + 23T^{2}$$
29 $$1 + 9.37T + 29T^{2}$$
31 $$1 - 2.91T + 31T^{2}$$
37 $$1 - 5.40T + 37T^{2}$$
41 $$1 - 8.45T + 41T^{2}$$
43 $$1 - 6.09T + 43T^{2}$$
47 $$1 - 13.2T + 47T^{2}$$
53 $$1 + 5.93T + 53T^{2}$$
59 $$1 - 8.45T + 59T^{2}$$
61 $$1 + 1.96T + 61T^{2}$$
71 $$1 - 5.61T + 71T^{2}$$
73 $$1 + 12.8T + 73T^{2}$$
79 $$1 - 11.8T + 79T^{2}$$
83 $$1 + 0.105T + 83T^{2}$$
89 $$1 - 16.2T + 89T^{2}$$
97 $$1 - 3.04T + 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

−7.79196754163964758584457197370, −7.29792573233465699145582275364, −6.44937470681751327200263546527, −5.91063647297601553801023757949, −5.38299390647734274583868840495, −4.37912011577070927089298769787, −3.60768236894160735178602150946, −2.82408507323631665595631495747, −2.23041256326920294202043202255, −0.72780526784434483781654196629, 0.72780526784434483781654196629, 2.23041256326920294202043202255, 2.82408507323631665595631495747, 3.60768236894160735178602150946, 4.37912011577070927089298769787, 5.38299390647734274583868840495, 5.91063647297601553801023757949, 6.44937470681751327200263546527, 7.29792573233465699145582275364, 7.79196754163964758584457197370