# Properties

 Degree 2 Conductor $2 \cdot 5 \cdot 73 \cdot 137$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + 2-s + 2·3-s + 4-s + 5-s + 2·6-s − 4·7-s + 8-s + 9-s + 10-s + 2·11-s + 2·12-s + 2·13-s − 4·14-s + 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 8·21-s + 2·22-s − 4·23-s + 2·24-s + 25-s + 2·26-s − 4·27-s − 4·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.577·12-s + 0.554·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.74·21-s + 0.426·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$100010$$    =    $$2 \cdot 5 \cdot 73 \cdot 137$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{100010} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 100010,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $7.117749770$ $L(\frac12)$ $\approx$ $7.117749770$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;73,\;137\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5,\;73,\;137\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - T$$
5 $$1 - T$$
73 $$1 + T$$
137 $$1 - T$$
good3 $$1 - 2 T + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 2 T + p T^{2}$$
71 $$1 - 12 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 - 18 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

−13.85282592859169, −13.35612205528751, −12.91729364152457, −12.47765213147414, −11.98752240145763, −11.39185343026111, −10.84070573562196, −10.04963189958537, −9.795241183554139, −9.309576093978146, −8.949589423919180, −8.214067948631447, −7.716652793500672, −7.241661026950722, −6.477704173319681, −6.054357320177607, −5.886456439561804, −4.938120686743562, −4.237564015115748, −3.678860108776172, −3.274054357151089, −2.764702243195555, −2.322131851465763, −1.416548322569268, −0.6945487919878361, 0.6945487919878361, 1.416548322569268, 2.322131851465763, 2.764702243195555, 3.274054357151089, 3.678860108776172, 4.237564015115748, 4.938120686743562, 5.886456439561804, 6.054357320177607, 6.477704173319681, 7.241661026950722, 7.716652793500672, 8.214067948631447, 8.949589423919180, 9.309576093978146, 9.795241183554139, 10.04963189958537, 10.84070573562196, 11.39185343026111, 11.98752240145763, 12.47765213147414, 12.91729364152457, 13.35612205528751, 13.85282592859169