Properties

 Degree 2 Conductor $3 \cdot 5^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 2·2-s + 3-s + 2·4-s − 2·6-s + 3·7-s + 9-s + 2·11-s + 2·12-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s − 5·19-s + 3·21-s − 4·22-s − 6·23-s + 2·26-s + 27-s + 6·28-s + 10·29-s − 3·31-s + 8·32-s + 2·33-s + 4·34-s + 2·36-s − 2·37-s + ⋯
 L(s)  = 1 − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s − 1.14·19-s + 0.654·21-s − 0.852·22-s − 1.25·23-s + 0.392·26-s + 0.192·27-s + 1.13·28-s + 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.348·33-s + 0.685·34-s + 1/3·36-s − 0.328·37-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$75$$    =    $$3 \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{75} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 75,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.6272349294$ $L(\frac12)$ $\approx$ $0.6272349294$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 - T$$
5 $$1$$
good2 $$1 + p T + p T^{2}$$
7 $$1 - 3 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 + T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 - 10 T + p T^{2}$$
31 $$1 + 3 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 8 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
53 $$1 - 4 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 - 7 T + p T^{2}$$
67 $$1 - 3 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 14 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 17 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

−19.54013625100329, −18.44712392748784, −17.69500123439249, −16.94587616486224, −15.75065958839187, −14.63360099431737, −13.71100770465531, −12.06535678251562, −10.88974094536361, −9.903619965777680, −8.655074250265875, −8.114757177440921, −6.772872512574206, −4.480180360053958, −1.903199556547081, 1.903199556547081, 4.480180360053958, 6.772872512574206, 8.114757177440921, 8.655074250265875, 9.903619965777680, 10.88974094536361, 12.06535678251562, 13.71100770465531, 14.63360099431737, 15.75065958839187, 16.94587616486224, 17.69500123439249, 18.44712392748784, 19.54013625100329