# Properties

 Label 2-13e2-13.12-c3-0-16 Degree $2$ Conductor $169$ Sign $-0.832 - 0.554i$ Analytic cond. $9.97132$ Root an. cond. $3.15774$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 4.56i·2-s + 8.68·3-s − 12.8·4-s − 2.80i·5-s + 39.6i·6-s + 9.56i·7-s − 21.9i·8-s + 48.4·9-s + 12.8·10-s + 39.4i·11-s − 111.·12-s − 43.6·14-s − 24.3i·15-s − 2.42·16-s − 2.01·17-s + 220. i·18-s + ⋯
 L(s)  = 1 + 1.61i·2-s + 1.67·3-s − 1.60·4-s − 0.251i·5-s + 2.69i·6-s + 0.516i·7-s − 0.969i·8-s + 1.79·9-s + 0.405·10-s + 1.08i·11-s − 2.67·12-s − 0.832·14-s − 0.419i·15-s − 0.0378·16-s − 0.0287·17-s + 2.89i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $-0.832 - 0.554i$ Analytic conductor: $$9.97132$$ Root analytic conductor: $$3.15774$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{169} (168, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 169,\ (\ :3/2),\ -0.832 - 0.554i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.782577 + 2.58467i$$ $$L(\frac12)$$ $$\approx$$ $$0.782577 + 2.58467i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1$$
good2 $$1 - 4.56iT - 8T^{2}$$
3 $$1 - 8.68T + 27T^{2}$$
5 $$1 + 2.80iT - 125T^{2}$$
7 $$1 - 9.56iT - 343T^{2}$$
11 $$1 - 39.4iT - 1.33e3T^{2}$$
17 $$1 + 2.01T + 4.91e3T^{2}$$
19 $$1 - 60.1iT - 6.85e3T^{2}$$
23 $$1 + 4.46T + 1.21e4T^{2}$$
29 $$1 - 140.T + 2.43e4T^{2}$$
31 $$1 + 136. iT - 2.97e4T^{2}$$
37 $$1 + 185. iT - 5.06e4T^{2}$$
41 $$1 + 310. iT - 6.89e4T^{2}$$
43 $$1 + 427.T + 7.95e4T^{2}$$
47 $$1 + 258. iT - 1.03e5T^{2}$$
53 $$1 - 612.T + 1.48e5T^{2}$$
59 $$1 + 517. iT - 2.05e5T^{2}$$
61 $$1 + 161.T + 2.26e5T^{2}$$
67 $$1 - 49.8iT - 3.00e5T^{2}$$
71 $$1 + 279. iT - 3.57e5T^{2}$$
73 $$1 - 467. iT - 3.89e5T^{2}$$
79 $$1 - 37.5T + 4.93e5T^{2}$$
83 $$1 - 76.1iT - 5.71e5T^{2}$$
89 $$1 - 202. iT - 7.04e5T^{2}$$
97 $$1 - 1.17e3iT - 9.12e5T^{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

−13.20939173019579476727431356775, −12.21451829364829241199856697944, −10.09290376998499119091692084964, −9.109873485649937062865630222199, −8.481263090631912848501989074009, −7.63247033710245601900340088075, −6.74878191153807085687488048469, −5.23193504129251305288830246955, −3.99540257240435191514380096063, −2.24598204701257982905979197878, 1.16928263155727204969620560755, 2.72962649891808148369347214836, 3.31982572340343047053478756121, 4.52076312914273031231803941619, 6.90367541249636820555335747385, 8.333306117364102748965948730040, 8.974202582625530949459316984411, 10.03411706426754121108194766680, 10.76943041222701393129096120205, 11.86284048019958281353605904986