Properties

 Label 2-2028-13.12-c3-0-22 Degree $2$ Conductor $2028$ Sign $-0.832 - 0.554i$ Analytic cond. $119.655$ Root an. cond. $10.9387$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + 3·3-s − 2i·5-s + 32i·7-s + 9·9-s + 68i·11-s − 6i·15-s + 14·17-s + 4i·19-s + 96i·21-s − 72·23-s + 121·25-s + 27·27-s + 102·29-s − 136i·31-s + 204i·33-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.178i·5-s + 1.72i·7-s + 0.333·9-s + 1.86i·11-s − 0.103i·15-s + 0.199·17-s + 0.0482i·19-s + 0.997i·21-s − 0.652·23-s + 0.967·25-s + 0.192·27-s + 0.653·29-s − 0.787i·31-s + 1.07i·33-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{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: $$2028$$    =    $$2^{2} \cdot 3 \cdot 13^{2}$$ Sign: $-0.832 - 0.554i$ Analytic conductor: $$119.655$$ Root analytic conductor: $$10.9387$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{2028} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2028,\ (\ :3/2),\ -0.832 - 0.554i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.307576987$$ $$L(\frac12)$$ $$\approx$$ $$2.307576987$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - 3T$$
13 $$1$$
good5 $$1 + 2iT - 125T^{2}$$
7 $$1 - 32iT - 343T^{2}$$
11 $$1 - 68iT - 1.33e3T^{2}$$
17 $$1 - 14T + 4.91e3T^{2}$$
19 $$1 - 4iT - 6.85e3T^{2}$$
23 $$1 + 72T + 1.21e4T^{2}$$
29 $$1 - 102T + 2.43e4T^{2}$$
31 $$1 + 136iT - 2.97e4T^{2}$$
37 $$1 - 386iT - 5.06e4T^{2}$$
41 $$1 - 250iT - 6.89e4T^{2}$$
43 $$1 - 140T + 7.95e4T^{2}$$
47 $$1 - 296iT - 1.03e5T^{2}$$
53 $$1 - 526T + 1.48e5T^{2}$$
59 $$1 + 332iT - 2.05e5T^{2}$$
61 $$1 + 410T + 2.26e5T^{2}$$
67 $$1 - 596iT - 3.00e5T^{2}$$
71 $$1 + 880iT - 3.57e5T^{2}$$
73 $$1 + 506iT - 3.89e5T^{2}$$
79 $$1 + 640T + 4.93e5T^{2}$$
83 $$1 - 1.38e3iT - 5.71e5T^{2}$$
89 $$1 + 1.45e3iT - 7.04e5T^{2}$$
97 $$1 + 446iT - 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

−9.124274325822656895203433760919, −8.374353947812951283536540023160, −7.71823997685475186750354013703, −6.75685636226184684741923616383, −5.96066261953728101922559712026, −4.94688989206704124506213762350, −4.39768572170333406033630841599, −2.99794688425899016482321795487, −2.33425926609340014814110981864, −1.48225111171507079653476834222, 0.44891388009551541113233175732, 1.16898147348237885581706773418, 2.64924450538313261881176546796, 3.60408325007648737536345349268, 4.02445682722962573778310429542, 5.23672851518401069917398648869, 6.22415003086038462786123877335, 7.06620956274989615360234939962, 7.64628119322417793580509004451, 8.519926980992739635935754671017