# Properties

 Degree 2 Conductor 29 Sign $-0.681 + 0.731i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−3.24 + 2.03i)2-s + (−2.23 − 0.780i)3-s + (4.63 − 9.61i)4-s + (−5.12 − 1.17i)5-s + (8.82 − 2.01i)6-s + (−6.56 + 3.16i)7-s + (2.86 + 25.4i)8-s + (−2.67 − 2.13i)9-s + (19.0 − 6.65i)10-s + (−0.480 + 4.26i)11-s + (−17.8 + 17.8i)12-s + (2.73 − 2.17i)13-s + (14.8 − 23.6i)14-s + (10.5 + 6.61i)15-s + (−34.4 − 43.2i)16-s + (−6.15 − 6.15i)17-s + ⋯
 L(s)  = 1 + (−1.62 + 1.01i)2-s + (−0.743 − 0.260i)3-s + (1.15 − 2.40i)4-s + (−1.02 − 0.234i)5-s + (1.47 − 0.335i)6-s + (−0.938 + 0.451i)7-s + (0.357 + 3.17i)8-s + (−0.296 − 0.236i)9-s + (1.90 − 0.665i)10-s + (−0.0437 + 0.387i)11-s + (−1.48 + 1.48i)12-s + (0.210 − 0.167i)13-s + (1.06 − 1.68i)14-s + (0.701 + 0.440i)15-s + (−2.15 − 2.70i)16-s + (−0.362 − 0.362i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$29$$ $$\varepsilon$$ = $-0.681 + 0.731i$ motivic weight = $$2$$ character : $\chi_{29} (10, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 29,\ (\ :1),\ -0.681 + 0.731i)$ $L(\frac{3}{2})$ $\approx$ $0.00916210 - 0.0210649i$ $L(\frac12)$ $\approx$ $0.00916210 - 0.0210649i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 29$, $$F_p$$ is a polynomial of degree 2. If $p = 29$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 $$1 + (0.661 + 28.9i)T$$
good2 $$1 + (3.24 - 2.03i)T + (1.73 - 3.60i)T^{2}$$
3 $$1 + (2.23 + 0.780i)T + (7.03 + 5.61i)T^{2}$$
5 $$1 + (5.12 + 1.17i)T + (22.5 + 10.8i)T^{2}$$
7 $$1 + (6.56 - 3.16i)T + (30.5 - 38.3i)T^{2}$$
11 $$1 + (0.480 - 4.26i)T + (-117. - 26.9i)T^{2}$$
13 $$1 + (-2.73 + 2.17i)T + (37.6 - 164. i)T^{2}$$
17 $$1 + (6.15 + 6.15i)T + 289iT^{2}$$
19 $$1 + (-5.18 - 14.8i)T + (-282. + 225. i)T^{2}$$
23 $$1 + (3.58 + 15.7i)T + (-476. + 229. i)T^{2}$$
31 $$1 + (34.6 - 21.8i)T + (416. - 865. i)T^{2}$$
37 $$1 + (5.62 + 49.9i)T + (-1.33e3 + 304. i)T^{2}$$
41 $$1 + (0.940 - 0.940i)T - 1.68e3iT^{2}$$
43 $$1 + (10.2 - 16.3i)T + (-802. - 1.66e3i)T^{2}$$
47 $$1 + (59.1 + 6.66i)T + (2.15e3 + 491. i)T^{2}$$
53 $$1 + (8.89 - 38.9i)T + (-2.53e3 - 1.21e3i)T^{2}$$
59 $$1 - 6.74T + 3.48e3T^{2}$$
61 $$1 + (-74.0 - 25.9i)T + (2.90e3 + 2.32e3i)T^{2}$$
67 $$1 + (-29.6 - 23.6i)T + (998. + 4.37e3i)T^{2}$$
71 $$1 + (-65.3 + 52.0i)T + (1.12e3 - 4.91e3i)T^{2}$$
73 $$1 + (38.0 + 23.8i)T + (2.31e3 + 4.80e3i)T^{2}$$
79 $$1 + (120. - 13.6i)T + (6.08e3 - 1.38e3i)T^{2}$$
83 $$1 + (-7.07 - 3.40i)T + (4.29e3 + 5.38e3i)T^{2}$$
89 $$1 + (94.3 - 59.3i)T + (3.43e3 - 7.13e3i)T^{2}$$
97 $$1 + (13.0 - 4.56i)T + (7.35e3 - 5.86e3i)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

−16.39893189425842696473278772728, −15.91168729542790061802049600618, −14.73763408081321576333734425535, −12.29813590574804372729865809783, −11.09351881550986896700829669709, −9.612823707609008283586474559180, −8.358859343762755575316241919745, −7.00249672070072389800738930879, −5.80355889069658188716128464523, −0.05084524685827240654317412974, 3.46634478136846994510818229595, 6.95912847978972738639899830936, 8.372370253764244689510847521424, 9.845489925486203494800165633379, 11.07048394703905421537377739180, 11.57638043757604854141600071158, 13.03513826772927490015549486029, 15.78458805813984800796246179087, 16.48315914365042584666768014251, 17.42744572115861841452773928672