# Properties

 Degree 2 Conductor 13 Sign $0.973 - 0.230i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (0.219 + 0.379i)2-s + (1.84 + 3.19i)3-s + (3.90 − 6.76i)4-s − 17.8·5-s + (−0.807 + 1.39i)6-s + (−2.71 + 4.70i)7-s + 6.93·8-s + (6.71 − 11.6i)9-s + (−3.90 − 6.76i)10-s + (11.2 + 19.4i)11-s + 28.7·12-s + (21.9 + 41.4i)13-s − 2.38·14-s + (−32.8 − 56.8i)15-s + (−29.7 − 51.4i)16-s + (−33.9 + 58.8i)17-s + ⋯
 L(s)  = 1 + (0.0775 + 0.134i)2-s + (0.354 + 0.614i)3-s + (0.487 − 0.845i)4-s − 1.59·5-s + (−0.0549 + 0.0951i)6-s + (−0.146 + 0.254i)7-s + 0.306·8-s + (0.248 − 0.430i)9-s + (−0.123 − 0.213i)10-s + (0.307 + 0.532i)11-s + 0.692·12-s + (0.468 + 0.883i)13-s − 0.0455·14-s + (−0.564 − 0.978i)15-s + (−0.464 − 0.804i)16-s + (−0.484 + 0.839i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.973 - 0.230i$ motivic weight = $$3$$ character : $\chi_{13} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :3/2),\ 0.973 - 0.230i)$ $L(2)$ $\approx$ $0.982027 + 0.114691i$ $L(\frac12)$ $\approx$ $0.982027 + 0.114691i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $$F_p$$ is a polynomial of degree 2. If $p = 13$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 $$1 + (-21.9 - 41.4i)T$$
good2 $$1 + (-0.219 - 0.379i)T + (-4 + 6.92i)T^{2}$$
3 $$1 + (-1.84 - 3.19i)T + (-13.5 + 23.3i)T^{2}$$
5 $$1 + 17.8T + 125T^{2}$$
7 $$1 + (2.71 - 4.70i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (-11.2 - 19.4i)T + (-665.5 + 1.15e3i)T^{2}$$
17 $$1 + (33.9 - 58.8i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-40.4 + 69.9i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (70.2 + 121. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (-53.3 - 92.3i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + 276.T + 2.97e4T^{2}$$
37 $$1 + (-2.14 - 3.71i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + (113. + 197. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (13.7 - 23.8i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 - 318.T + 1.03e5T^{2}$$
53 $$1 + 67.6T + 1.48e5T^{2}$$
59 $$1 + (-145. + 252. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (331. - 574. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-212. - 368. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + (-76.4 + 132. i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 - 117.T + 3.89e5T^{2}$$
79 $$1 - 202.T + 4.93e5T^{2}$$
83 $$1 - 336.T + 5.71e5T^{2}$$
89 $$1 + (359. + 621. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + (379. - 657. i)T + (-4.56e5 - 7.90e5i)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

−19.74589105824024445259505976782, −18.56744771117491312836050583330, −16.18066007274649173202413446868, −15.44544303217164622985200394439, −14.55847462133591598399845749321, −12.14658068558979807099510010550, −10.80163656029455800316775363706, −9.009732338948342568151131608751, −6.90403086611684217568404125648, −4.19022483144951431422246083571, 3.58625571942394858965526178874, 7.34663469630280678944139989589, 8.162302933922337311779364217199, 11.12718488914698104803801456386, 12.24656201119802978584149472660, 13.55961242981874110454866407054, 15.60674160626015570317270278982, 16.44675986116577346436403731362, 18.35101931534678645970805260435, 19.67780105452462756970465910990