# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 23 \cdot 29$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 + 2-s − 3-s − 4-s + 5-s − 6-s − 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s − 15-s − 16-s − 6·17-s + 18-s + 4·19-s − 20-s + 4·22-s − 23-s + 3·24-s + 25-s − 2·26-s − 27-s + 29-s − 30-s + 4·31-s + 5·32-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.185·29-s − 0.182·30-s + 0.718·31-s + 0.883·32-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$10005$$    =    $$3 \cdot 5 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{10005} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 10005,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;23,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + T$$
5 $$1 - T$$
23 $$1 + T$$
29 $$1 - T$$
good2 $$1 - T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 6 T + p T^{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

−17.17718612691170, −16.30636475088869, −15.73588379599894, −15.09616656410795, −14.50618449612901, −14.01570979062440, −13.46538500862401, −13.00211296240947, −12.27194063929149, −11.80240693551841, −11.37018780481208, −10.49094406420653, −9.768763226507564, −9.338643346579857, −8.768821891142581, −7.992315866950088, −6.946005974467450, −6.489899452271880, −5.962355840346338, −5.032412579610391, −4.749854360805160, −3.949153321139294, −3.229960703715178, −2.218764488273508, −1.180916365052916, 0, 1.180916365052916, 2.218764488273508, 3.229960703715178, 3.949153321139294, 4.749854360805160, 5.032412579610391, 5.962355840346338, 6.489899452271880, 6.946005974467450, 7.992315866950088, 8.768821891142581, 9.338643346579857, 9.768763226507564, 10.49094406420653, 11.37018780481208, 11.80240693551841, 12.27194063929149, 13.00211296240947, 13.46538500862401, 14.01570979062440, 14.50618449612901, 15.09616656410795, 15.73588379599894, 16.30636475088869, 17.17718612691170