Properties

 Degree 2 Conductor $2^{7} \cdot 19$ Sign $-i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯
 L(s)  = 1 + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$2432$$    =    $$2^{7} \cdot 19$$ $$\varepsilon$$ = $-i$ motivic weight = $$0$$ character : $\chi_{2432} (1025, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2432,\ (\ :0),\ -i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.394108835$$ $$L(\frac12)$$ $$\approx$$ $$1.394108835$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;19\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
19 $$1 - T$$
good3 $$1 - 1.41iT - T^{2}$$
5 $$1 - T + T^{2}$$
7 $$1 + T + T^{2}$$
11 $$1 - T + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 - T + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + 1.41iT - T^{2}$$
31 $$1 - 1.41iT - T^{2}$$
37 $$1 - 1.41iT - T^{2}$$
41 $$1 - 1.41iT - T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + T + T^{2}$$
53 $$1 + 1.41iT - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 + 1.41iT - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + T + T^{2}$$
79 $$1 + 1.41iT - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + 1.41iT - T^{2}$$
97 $$1 - 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

−9.684980502553304155216705643809, −8.922507338867491402626439943921, −7.968873330144470489835380398063, −6.70288708059089294993608278330, −6.19056472651110000227843988914, −5.31217614313115425253310201074, −4.62759204067765548397395609711, −3.46400379384153039385833720929, −3.13738320677443425497528763966, −1.53318185159870885012074061222, 1.06602819575941458445746130751, 1.93331361085573211956091496664, 2.96023488152114112165797315315, 3.89527742979712287928193362844, 5.52831608313672084366181938018, 5.86054244901229944584324310744, 6.78974174744660007977565909557, 7.13590312465581883611692345281, 8.037016542285944529528452076250, 9.046714716080296934919855591226