Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $0.998 - 0.0618i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 5.26i·11-s + (3.16 − 3.16i)13-s + 1.00·14-s − 1.00·16-s + (−3.05 + 3.05i)17-s + (3.72 + 3.72i)22-s + (−4.32 − 4.32i)23-s − 4.46i·26-s + (0.707 − 0.707i)28-s + 9.96·29-s + 1.26·31-s + (−0.707 + 0.707i)32-s + ⋯
 L(s)  = 1 + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 1.58i·11-s + (0.876 − 0.876i)13-s + 0.267·14-s − 0.250·16-s + (−0.741 + 0.741i)17-s + (0.793 + 0.793i)22-s + (−0.901 − 0.901i)23-s − 0.876i·26-s + (0.133 − 0.133i)28-s + 1.84·29-s + 0.227·31-s + (−0.125 + 0.125i)32-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.998 - 0.0618i$ motivic weight = $$1$$ character : $\chi_{3150} (1457, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ 0.998 - 0.0618i)$ $L(1)$ $\approx$ $2.455910178$ $L(\frac12)$ $\approx$ $2.455910178$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.707 + 0.707i)T$$
3 $$1$$
5 $$1$$
7 $$1 + (-0.707 - 0.707i)T$$
good11 $$1 - 5.26iT - 11T^{2}$$
13 $$1 + (-3.16 + 3.16i)T - 13iT^{2}$$
17 $$1 + (3.05 - 3.05i)T - 17iT^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + (4.32 + 4.32i)T + 23iT^{2}$$
29 $$1 - 9.96T + 29T^{2}$$
31 $$1 - 1.26T + 31T^{2}$$
37 $$1 + (-2.93 - 2.93i)T + 37iT^{2}$$
41 $$1 - 10.6iT - 41T^{2}$$
43 $$1 + (0.597 - 0.597i)T - 43iT^{2}$$
47 $$1 + (-3.42 + 3.42i)T - 47iT^{2}$$
53 $$1 + (-9.88 - 9.88i)T + 53iT^{2}$$
59 $$1 - 3.12T + 59T^{2}$$
61 $$1 - 3.05T + 61T^{2}$$
67 $$1 + (4.59 + 4.59i)T + 67iT^{2}$$
71 $$1 - 9.23iT - 71T^{2}$$
73 $$1 + (10.2 - 10.2i)T - 73iT^{2}$$
79 $$1 - 3.57iT - 79T^{2}$$
83 $$1 + (-6.21 - 6.21i)T + 83iT^{2}$$
89 $$1 + 3.61T + 89T^{2}$$
97 $$1 + (-4.63 - 4.63i)T + 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}