# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 7 \cdot 601$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 6·19-s − 20-s + 21-s − 2·22-s + 3·23-s + 24-s − 4·25-s + 27-s + 28-s + 2·29-s − 30-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 25242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 25242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$25242$$    =    $$2 \cdot 3 \cdot 7 \cdot 601$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{25242} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 25242,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $5.196725424$ $L(\frac12)$ $\approx$ $5.196725424$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7,\;601\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;7,\;601\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 - T$$
7 $$1 - T$$
601 $$1 + T$$
good5 $$1 + T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
23 $$1 - 3 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 9 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 9 T + p T^{2}$$
59 $$1 + 3 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 + 16 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 + 11 T + p T^{2}$$
83 $$1 + 7 T + p T^{2}$$
89 $$1 - 13 T + p T^{2}$$
97 $$1 - 7 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

−15.40627011346761, −14.59847198644234, −14.39543095996447, −13.79771911184653, −13.25391589935274, −12.79813996276828, −12.10730009542352, −11.65982649581201, −11.24568768843637, −10.37509818390062, −10.04934139734870, −9.287973003190630, −8.712428078195466, −7.827859495620335, −7.614050244456849, −7.249742908418365, −6.077888643095739, −5.790384031496747, −4.903327151680186, −4.470047242745831, −3.748243635203316, −2.959529541128039, −2.699408847650457, −1.568774578473078, −0.8198843654055304, 0.8198843654055304, 1.568774578473078, 2.699408847650457, 2.959529541128039, 3.748243635203316, 4.470047242745831, 4.903327151680186, 5.790384031496747, 6.077888643095739, 7.249742908418365, 7.614050244456849, 7.827859495620335, 8.712428078195466, 9.287973003190630, 10.04934139734870, 10.37509818390062, 11.24568768843637, 11.65982649581201, 12.10730009542352, 12.79813996276828, 13.25391589935274, 13.79771911184653, 14.39543095996447, 14.59847198644234, 15.40627011346761