# Properties

 Label 2-2e4-4.3-c8-0-3 Degree $2$ Conductor $16$ Sign $-1$ Analytic cond. $6.51805$ Root an. cond. $2.55304$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 141. i·3-s − 510·5-s + 2.55e3i·7-s − 1.35e4·9-s − 1.91e4i·11-s − 2.77e4·13-s + 7.24e4i·15-s + 5.03e4·17-s − 1.08e5i·19-s + 3.62e5·21-s − 1.76e5i·23-s − 1.30e5·25-s + 9.99e5i·27-s + 5.49e4·29-s − 1.17e6i·31-s + ⋯
 L(s)  = 1 − 1.75i·3-s − 0.816·5-s + 1.06i·7-s − 2.07·9-s − 1.30i·11-s − 0.970·13-s + 1.43i·15-s + 0.603·17-s − 0.833i·19-s + 1.86·21-s − 0.630i·23-s − 0.334·25-s + 1.88i·27-s + 0.0777·29-s − 1.27i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$16$$    =    $$2^{4}$$ Sign: $-1$ Analytic conductor: $$6.51805$$ Root analytic conductor: $$2.55304$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{16} (15, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 16,\ (\ :4),\ -1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$-0.800545i$$ $$L(\frac12)$$ $$\approx$$ $$-0.800545i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 141. iT - 6.56e3T^{2}$$
5 $$1 + 510T + 3.90e5T^{2}$$
7 $$1 - 2.55e3iT - 5.76e6T^{2}$$
11 $$1 + 1.91e4iT - 2.14e8T^{2}$$
13 $$1 + 2.77e4T + 8.15e8T^{2}$$
17 $$1 - 5.03e4T + 6.97e9T^{2}$$
19 $$1 + 1.08e5iT - 1.69e10T^{2}$$
23 $$1 + 1.76e5iT - 7.83e10T^{2}$$
29 $$1 - 5.49e4T + 5.00e11T^{2}$$
31 $$1 + 1.17e6iT - 8.52e11T^{2}$$
37 $$1 - 7.93e5T + 3.51e12T^{2}$$
41 $$1 + 7.55e4T + 7.98e12T^{2}$$
43 $$1 + 4.99e5iT - 1.16e13T^{2}$$
47 $$1 + 2.86e6iT - 2.38e13T^{2}$$
53 $$1 - 1.11e7T + 6.22e13T^{2}$$
59 $$1 - 2.18e7iT - 1.46e14T^{2}$$
61 $$1 + 2.38e7T + 1.91e14T^{2}$$
67 $$1 + 7.49e6iT - 4.06e14T^{2}$$
71 $$1 - 1.00e7iT - 6.45e14T^{2}$$
73 $$1 - 6.51e6T + 8.06e14T^{2}$$
79 $$1 - 4.87e7iT - 1.51e15T^{2}$$
83 $$1 + 7.34e7iT - 2.25e15T^{2}$$
89 $$1 - 8.67e7T + 3.93e15T^{2}$$
97 $$1 + 4.66e7T + 7.83e15T^{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

−16.76646688473833333705190776647, −15.04648351514145639173738149473, −13.58529294497711327043524366998, −12.27801924036190834159341145690, −11.53329504294888990292621947213, −8.645736225117550084400133122170, −7.49769515905505937595628963416, −5.88899677477203516236891189083, −2.60379807457954539457520050565, −0.43933865474779621515255564421, 3.70734191623645052316818471976, 4.81358410003335607058665818201, 7.62006934548952151741623958571, 9.649148753062120836408863498092, 10.51838123367464733612617225198, 12.05418020255978570190492316653, 14.35818415366575748952044509475, 15.30052339723711556560907967474, 16.40495392749809605101946713050, 17.38624172233306483745881687732