Properties

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

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Dirichlet series

 L(s)  = 1 − 4·2-s − 26·3-s + 16·4-s − 25·5-s + 104·6-s − 22·7-s − 64·8-s + 433·9-s + 100·10-s − 768·11-s − 416·12-s − 46·13-s + 88·14-s + 650·15-s + 256·16-s + 378·17-s − 1.73e3·18-s + 1.10e3·19-s − 400·20-s + 572·21-s + 3.07e3·22-s − 1.98e3·23-s + 1.66e3·24-s + 625·25-s + 184·26-s − 4.94e3·27-s − 352·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.66·3-s + 1/2·4-s − 0.447·5-s + 1.17·6-s − 0.169·7-s − 0.353·8-s + 1.78·9-s + 0.316·10-s − 1.91·11-s − 0.833·12-s − 0.0754·13-s + 0.119·14-s + 0.745·15-s + 1/4·16-s + 0.317·17-s − 1.25·18-s + 0.699·19-s − 0.223·20-s + 0.283·21-s + 1.35·22-s − 0.782·23-s + 0.589·24-s + 1/5·25-s + 0.0533·26-s − 1.30·27-s − 0.0848·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$10$$    =    $$2 \cdot 5$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{10} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 10,\ (\ :5/2),\ -1)$ $L(3)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{7}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + p^{2} T$$
5 $$1 + p^{2} T$$
good3 $$1 + 26 T + p^{5} T^{2}$$
7 $$1 + 22 T + p^{5} T^{2}$$
11 $$1 + 768 T + p^{5} T^{2}$$
13 $$1 + 46 T + p^{5} T^{2}$$
17 $$1 - 378 T + p^{5} T^{2}$$
19 $$1 - 1100 T + p^{5} T^{2}$$
23 $$1 + 1986 T + p^{5} T^{2}$$
29 $$1 + 5610 T + p^{5} T^{2}$$
31 $$1 + 3988 T + p^{5} T^{2}$$
37 $$1 + 142 T + p^{5} T^{2}$$
41 $$1 - 1542 T + p^{5} T^{2}$$
43 $$1 + 5026 T + p^{5} T^{2}$$
47 $$1 - 24738 T + p^{5} T^{2}$$
53 $$1 + 14166 T + p^{5} T^{2}$$
59 $$1 - 28380 T + p^{5} T^{2}$$
61 $$1 - 5522 T + p^{5} T^{2}$$
67 $$1 + 24742 T + p^{5} T^{2}$$
71 $$1 - 42372 T + p^{5} T^{2}$$
73 $$1 + 52126 T + p^{5} T^{2}$$
79 $$1 + 39640 T + p^{5} T^{2}$$
83 $$1 + 59826 T + p^{5} T^{2}$$
89 $$1 - 57690 T + p^{5} T^{2}$$
97 $$1 + 144382 T + p^{5} 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

−18.71089443975698703059680487653, −17.90753533391967973639010814268, −16.50594414723260966299811955003, −15.67204906179363141411647710651, −12.71607735457145862734525391353, −11.33655765402030971118823252433, −10.19557438841200567182587153987, −7.52312938746834257699973811858, −5.52108618060976421966780041472, 0, 5.52108618060976421966780041472, 7.52312938746834257699973811858, 10.19557438841200567182587153987, 11.33655765402030971118823252433, 12.71607735457145862734525391353, 15.67204906179363141411647710651, 16.50594414723260966299811955003, 17.90753533391967973639010814268, 18.71089443975698703059680487653