# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 59 \cdot 113$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s + 9-s + 2·10-s + 4·11-s − 2·12-s − 4·13-s + 2·14-s + 15-s − 4·16-s + 2·17-s − 2·18-s − 2·20-s + 21-s − 8·22-s − 7·23-s + 25-s + 8·26-s − 27-s − 2·28-s − 6·29-s − 2·30-s − 8·31-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.447·20-s + 0.218·21-s − 1.70·22-s − 1.45·23-s + 1/5·25-s + 1.56·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s − 1.43·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$100005$$    =    $$3 \cdot 5 \cdot 59 \cdot 113$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{100005} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(2,\ 100005,\ (\ :1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;59,\;113\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;59,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + T$$
5 $$1 + T$$
59 $$1 - T$$
113 $$1 + T$$
good2 $$1 + p T + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + 7 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 9 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 13 T + p T^{2}$$
71 $$1 - 16 T + p T^{2}$$
73 $$1 + T + p T^{2}$$
79 $$1 + 12 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 2 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

−14.35531784025453, −13.86292662999597, −13.11082212247720, −12.52710278025528, −12.16367166396146, −11.65144926239270, −11.28313252012876, −10.72279977632798, −10.15566873713051, −9.668672872282089, −9.517621128468811, −8.823375599317415, −8.317318598305679, −7.783200868260622, −7.226627710353203, −6.898114304772559, −6.423396267049531, −5.616533958778447, −5.153335960748824, −4.406843327480296, −3.767211558134807, −3.385658597216329, −2.172864560437771, −1.816414789089624, −1.070281893583117, 0, 0, 1.070281893583117, 1.816414789089624, 2.172864560437771, 3.385658597216329, 3.767211558134807, 4.406843327480296, 5.153335960748824, 5.616533958778447, 6.423396267049531, 6.898114304772559, 7.226627710353203, 7.783200868260622, 8.317318598305679, 8.823375599317415, 9.517621128468811, 9.668672872282089, 10.15566873713051, 10.72279977632798, 11.28313252012876, 11.65144926239270, 12.16367166396146, 12.52710278025528, 13.11082212247720, 13.86292662999597, 14.35531784025453