# Properties

 Label 2-896-224.13-c0-0-0 Degree $2$ Conductor $896$ Sign $0.555 + 0.831i$ Analytic cond. $0.447162$ Root an. cond. $0.668701$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (0.707 − 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (−0.292 − 0.707i)53-s − 1.00·63-s + (−0.292 + 0.707i)67-s + (−1.70 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (0.707 − 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (−0.292 − 0.707i)53-s − 1.00·63-s + (−0.292 + 0.707i)67-s + (−1.70 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$896$$    =    $$2^{7} \cdot 7$$ Sign: $0.555 + 0.831i$ Analytic conductor: $$0.447162$$ Root analytic conductor: $$0.668701$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{896} (657, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 896,\ (\ :0),\ 0.555 + 0.831i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9684709581$$ $$L(\frac12)$$ $$\approx$$ $$0.9684709581$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (-0.707 + 0.707i)T$$
good3 $$1 + (0.707 + 0.707i)T^{2}$$
5 $$1 + (-0.707 + 0.707i)T^{2}$$
11 $$1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}$$
13 $$1 + (-0.707 - 0.707i)T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + (-0.707 - 0.707i)T^{2}$$
23 $$1 + (-1 - i)T + iT^{2}$$
29 $$1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2}$$
41 $$1 - iT^{2}$$
43 $$1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2}$$
59 $$1 + (-0.707 + 0.707i)T^{2}$$
61 $$1 + (0.707 + 0.707i)T^{2}$$
67 $$1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2}$$
71 $$1 - iT^{2}$$
73 $$1 - iT^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 + (-0.707 - 0.707i)T^{2}$$
89 $$1 + iT^{2}$$
97 $$1 - T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.37867037697018834474307957650, −9.161401084981648883275826620811, −8.543719113023377707721609270467, −7.75667050804081421545669569929, −6.77820815604410505503678097931, −5.74965680322947126483517689282, −5.03038666312973927882021152941, −3.67321395568854535224913615701, −2.90007727478073064167155416647, −1.02917740701153812391221961772, 2.00195672710065480054624641733, 2.75027351849633254106181210761, 4.51445647198220240170724483400, 5.05266164470524463309420157174, 5.99864746304311802393021972678, 7.26823706403039517249601387275, 7.87941309732915751147963891809, 8.755953910041500744601541054870, 9.570677947728051515882716116117, 10.54301364278793173580832314525