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 + 6.36i·11-s + (1.69 + 1.69i)13-s − 1.00·14-s − 1.00·16-s + (−0.979 − 0.979i)17-s + (−4.49 + 4.49i)22-s + (1.38 − 1.38i)23-s + 2.39i·26-s + (−0.707 − 0.707i)28-s − 5.81·29-s + 2.36·31-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 + 1.91i·11-s + (0.469 + 0.469i)13-s − 0.267·14-s − 0.250·16-s + (−0.237 − 0.237i)17-s + (−0.959 + 0.959i)22-s + (0.288 − 0.288i)23-s + 0.469i·26-s + (−0.133 − 0.133i)28-s − 1.08·29-s + 0.424·31-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 : $\chi_{3150} (2843, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ -0.998 - 0.0618i)$ $L(1)$ $\approx$ $1.527882631$ $L(\frac12)$ $\approx$ $1.527882631$ $L(\frac{3}{2})$ 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 - 6.36iT - 11T^{2}$$
13 $$1 + (-1.69 - 1.69i)T + 13iT^{2}$$
17 $$1 + (0.979 + 0.979i)T + 17iT^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + (-1.38 + 1.38i)T - 23iT^{2}$$
29 $$1 + 5.81T + 29T^{2}$$
31 $$1 - 2.36T + 31T^{2}$$
37 $$1 + (0.157 - 0.157i)T - 37iT^{2}$$
41 $$1 - 8.94iT - 41T^{2}$$
43 $$1 + (5.88 + 5.88i)T + 43iT^{2}$$
47 $$1 + (3.05 + 3.05i)T + 47iT^{2}$$
53 $$1 + (0.192 - 0.192i)T - 53iT^{2}$$
59 $$1 - 10.3T + 59T^{2}$$
61 $$1 + 0.979T + 61T^{2}$$
67 $$1 + (9.88 - 9.88i)T - 67iT^{2}$$
71 $$1 + 12.3iT - 71T^{2}$$
73 $$1 + (4.71 + 4.71i)T + 73iT^{2}$$
79 $$1 - 6.68iT - 79T^{2}$$
83 $$1 + (11.3 - 11.3i)T - 83iT^{2}$$
89 $$1 + 11.8T + 89T^{2}$$
97 $$1 + (-9.39 + 9.39i)T - 97iT^{2}$$
show less
\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

−8.997953168729009525212100117108, −8.221915661676878484168935332709, −7.28068147534567633367499586131, −6.87775136827822296247540039018, −6.09376516280642384009845611390, −5.13155162471317596276817384209, −4.51802361562843326412430100848, −3.73093390634685240299064505242, −2.60741457284700853663918369216, −1.69074683003654057948147483787, 0.38493884641783686940225514723, 1.50078223723858994580341304764, 2.88299047034554229672844615564, 3.44318323232005031188363056223, 4.20382489190269773402663113128, 5.37508497514264324980789053002, 5.87737869227204390068919823285, 6.59373826726488548006048449025, 7.58859316447461786757448219537, 8.490565478262868409094037698099