# Properties

 Label 2-13e2-13.12-c1-0-2 Degree $2$ Conductor $169$ Sign $0.691 - 0.722i$ Analytic cond. $1.34947$ Root an. cond. $1.16166$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 0.554i·2-s + 0.801·3-s + 1.69·4-s + 2.80i·5-s + 0.445i·6-s − 2.69i·7-s + 2.04i·8-s − 2.35·9-s − 1.55·10-s − 1.19i·11-s + 1.35·12-s + 1.49·14-s + 2.24i·15-s + 2.24·16-s − 1.13·17-s − 1.30i·18-s + ⋯
 L(s)  = 1 + 0.392i·2-s + 0.462·3-s + 0.846·4-s + 1.25i·5-s + 0.181i·6-s − 1.01i·7-s + 0.724i·8-s − 0.785·9-s − 0.491·10-s − 0.361i·11-s + 0.391·12-s + 0.399·14-s + 0.580i·15-s + 0.561·16-s − 0.275·17-s − 0.308i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $0.691 - 0.722i$ Analytic conductor: $$1.34947$$ Root analytic conductor: $$1.16166$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{169} (168, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 169,\ (\ :1/2),\ 0.691 - 0.722i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.35538 + 0.578613i$$ $$L(\frac12)$$ $$\approx$$ $$1.35538 + 0.578613i$$ $$L(\frac{3}{2})$$ 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 - 0.554iT - 2T^{2}$$
3 $$1 - 0.801T + 3T^{2}$$
5 $$1 - 2.80iT - 5T^{2}$$
7 $$1 + 2.69iT - 7T^{2}$$
11 $$1 + 1.19iT - 11T^{2}$$
17 $$1 + 1.13T + 17T^{2}$$
19 $$1 + 1.93iT - 19T^{2}$$
23 $$1 - 4.60T + 23T^{2}$$
29 $$1 + 7.89T + 29T^{2}$$
31 $$1 + 5.89iT - 31T^{2}$$
37 $$1 - 0.951iT - 37T^{2}$$
41 $$1 + 3.31iT - 41T^{2}$$
43 $$1 + 7.15T + 43T^{2}$$
47 $$1 + 7.69iT - 47T^{2}$$
53 $$1 - 5.87T + 53T^{2}$$
59 $$1 + 0.0120iT - 59T^{2}$$
61 $$1 + 8.03T + 61T^{2}$$
67 $$1 - 9.25iT - 67T^{2}$$
71 $$1 - 13.7iT - 71T^{2}$$
73 $$1 - 12.8iT - 73T^{2}$$
79 $$1 - 0.807T + 79T^{2}$$
83 $$1 - 16.3iT - 83T^{2}$$
89 $$1 + 14.7iT - 89T^{2}$$
97 $$1 + 3.13iT - 97T^{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

−13.23172843026177316489265010637, −11.42699065598586314950194827702, −11.10911071766157646178444943363, −10.10481225846461631473521577550, −8.614244742138536684828859967988, −7.41409192544279426129151621974, −6.86912401101177441015696700849, −5.67399485917868166993388689587, −3.58631967682433352224547082499, −2.50507082103092567003728858686, 1.85363386397674879286724229876, 3.21612061595007245431824799924, 5.02991880792359541495104972319, 6.12411235041285051012484329717, 7.66471625855674818277110366204, 8.778823233459227880407182999975, 9.375482347468606611056271135866, 10.83927107409341649636334428678, 11.86302020321111432584732948672, 12.46411806192441046530082690304