Properties

Label 2-380-20.19-c2-0-100
Degree $2$
Conductor $380$
Sign $0.649 + 0.760i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.75 − 0.968i)2-s + 4.24·3-s + (2.12 − 3.38i)4-s + (4.77 + 1.49i)5-s + (7.43 − 4.11i)6-s − 7.86·7-s + (0.437 − 7.98i)8-s + 9.03·9-s + (9.79 − 2.00i)10-s − 2.63i·11-s + (9.02 − 14.3i)12-s − 5.11i·13-s + (−13.7 + 7.61i)14-s + (20.2 + 6.34i)15-s + (−6.96 − 14.4i)16-s + 12.2i·17-s + ⋯
L(s)  = 1  + (0.875 − 0.484i)2-s + 1.41·3-s + (0.531 − 0.847i)4-s + (0.954 + 0.298i)5-s + (1.23 − 0.685i)6-s − 1.12·7-s + (0.0547 − 0.998i)8-s + 1.00·9-s + (0.979 − 0.200i)10-s − 0.239i·11-s + (0.752 − 1.19i)12-s − 0.393i·13-s + (−0.982 + 0.543i)14-s + (1.35 + 0.423i)15-s + (−0.435 − 0.900i)16-s + 0.721i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.649 + 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.10143 - 1.88978i\)
\(L(\frac12)\) \(\approx\) \(4.10143 - 1.88978i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 + 0.968i)T \)
5 \( 1 + (-4.77 - 1.49i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 4.24T + 9T^{2} \)
7 \( 1 + 7.86T + 49T^{2} \)
11 \( 1 + 2.63iT - 121T^{2} \)
13 \( 1 + 5.11iT - 169T^{2} \)
17 \( 1 - 12.2iT - 289T^{2} \)
23 \( 1 - 17.2T + 529T^{2} \)
29 \( 1 - 4.95T + 841T^{2} \)
31 \( 1 - 59.4iT - 961T^{2} \)
37 \( 1 - 47.8iT - 1.36e3T^{2} \)
41 \( 1 + 4.38T + 1.68e3T^{2} \)
43 \( 1 + 51.2T + 1.84e3T^{2} \)
47 \( 1 + 41.3T + 2.20e3T^{2} \)
53 \( 1 - 42.9iT - 2.80e3T^{2} \)
59 \( 1 + 103. iT - 3.48e3T^{2} \)
61 \( 1 - 68.7T + 3.72e3T^{2} \)
67 \( 1 + 69.7T + 4.48e3T^{2} \)
71 \( 1 + 55.1iT - 5.04e3T^{2} \)
73 \( 1 + 45.6iT - 5.32e3T^{2} \)
79 \( 1 + 84.1iT - 6.24e3T^{2} \)
83 \( 1 - 2.16T + 6.88e3T^{2} \)
89 \( 1 - 89.6T + 7.92e3T^{2} \)
97 \( 1 - 36.5iT - 9.40e3T^{2} \)
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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

−10.82163762331308196118888119860, −10.05523569686158303903340956481, −9.400534704211203827203054692562, −8.445996594117371197252931423790, −6.94250634929294778255147183835, −6.24748806092967015151619247389, −4.98456271455276486899773610873, −3.29521287125732733206907215916, −3.05079948450911375372645779036, −1.68821255566400609313423340962, 2.18312646169271116799328190342, 3.03256349466803904483096075511, 4.12877937009491822367267462140, 5.45937848354214356190895719010, 6.52668736760412028786702747639, 7.37605071162331751205692170660, 8.518779803276429432018757056366, 9.322140410273127076793673739911, 9.971288588796229813213352815967, 11.49753433765137183986251073123

Graph of the $Z$-function along the critical line