# Properties

 Degree 2 Conductor $5^{2} \cdot 7 \cdot 23$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + 2-s − 4-s + 7-s − 3·8-s − 3·9-s + 2·11-s − 4·13-s + 14-s − 16-s + 6·17-s − 3·18-s − 8·19-s + 2·22-s − 23-s − 4·26-s − 28-s + 10·29-s + 10·31-s + 5·32-s + 6·34-s + 3·36-s − 8·37-s − 8·38-s − 2·41-s − 2·44-s − 46-s − 12·47-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s + 0.603·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.83·19-s + 0.426·22-s − 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.85·29-s + 1.79·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.31·37-s − 1.29·38-s − 0.312·41-s − 0.301·44-s − 0.147·46-s − 1.75·47-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4025$$    =    $$5^{2} \cdot 7 \cdot 23$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4025} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4025,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.793527565$ $L(\frac12)$ $\approx$ $1.793527565$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;7,\;23\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1$$
7 $$1 - T$$
23 $$1 + T$$
good2 $$1 - T + p T^{2}$$
3 $$1 + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + 8 T + p T^{2}$$
29 $$1 - 10 T + p T^{2}$$
31 $$1 - 10 T + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 - 4 T + p T^{2}$$
59 $$1 - 14 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 - 6 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 10 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

−17.89902136835414, −17.33010039214084, −17.15020474005605, −16.33852498998087, −15.32805347886893, −14.80598316001591, −14.28122462306671, −14.04072165108404, −13.18303095240544, −12.31889401363248, −12.05241851932183, −11.50622795161756, −10.27818074074263, −10.03209316112717, −8.983335752854459, −8.393376404030028, −7.970958775519788, −6.612458624116954, −6.257527422721416, −5.152608681437976, −4.865865592308691, −3.908898144464231, −3.095142491921366, −2.244196511532632, −0.6753660668505041, 0.6753660668505041, 2.244196511532632, 3.095142491921366, 3.908898144464231, 4.865865592308691, 5.152608681437976, 6.257527422721416, 6.612458624116954, 7.970958775519788, 8.393376404030028, 8.983335752854459, 10.03209316112717, 10.27818074074263, 11.50622795161756, 12.05241851932183, 12.31889401363248, 13.18303095240544, 14.04072165108404, 14.28122462306671, 14.80598316001591, 15.32805347886893, 16.33852498998087, 17.15020474005605, 17.33010039214084, 17.89902136835414