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Properties

Label 2-1620-15.14-c2-0-18
Degree $2$
Conductor $1620$
Sign $0.565 - 0.824i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (2.82 − 4.12i)5-s + 7.52i·7-s − 6.67i·11-s + 15.5i·13-s + 18.9·17-s − 27.1·19-s + 19.6·23-s + (−9.02 − 23.3i)25-s + 6.97i·29-s + 29.5·31-s + (31.0 + 21.2i)35-s + 52.2i·37-s + 60.8i·41-s + 10.6i·43-s − 75.8·47-s + ⋯
L(s)  = 1  + (0.565 − 0.824i)5-s + 1.07i·7-s − 0.606i·11-s + 1.19i·13-s + 1.11·17-s − 1.42·19-s + 0.852·23-s + (−0.361 − 0.932i)25-s + 0.240i·29-s + 0.952·31-s + (0.887 + 0.607i)35-s + 1.41i·37-s + 1.48i·41-s + 0.247i·43-s − 1.61·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.565 - 0.824i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.014337575\)
\(L(\frac12)\) \(\approx\) \(2.014337575\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.82 + 4.12i)T \)
good7 \( 1 - 7.52iT - 49T^{2} \)
11 \( 1 + 6.67iT - 121T^{2} \)
13 \( 1 - 15.5iT - 169T^{2} \)
17 \( 1 - 18.9T + 289T^{2} \)
19 \( 1 + 27.1T + 361T^{2} \)
23 \( 1 - 19.6T + 529T^{2} \)
29 \( 1 - 6.97iT - 841T^{2} \)
31 \( 1 - 29.5T + 961T^{2} \)
37 \( 1 - 52.2iT - 1.36e3T^{2} \)
41 \( 1 - 60.8iT - 1.68e3T^{2} \)
43 \( 1 - 10.6iT - 1.84e3T^{2} \)
47 \( 1 + 75.8T + 2.20e3T^{2} \)
53 \( 1 + 47.3T + 2.80e3T^{2} \)
59 \( 1 + 24.4iT - 3.48e3T^{2} \)
61 \( 1 - 40.7T + 3.72e3T^{2} \)
67 \( 1 + 58.9iT - 4.48e3T^{2} \)
71 \( 1 - 33.9iT - 5.04e3T^{2} \)
73 \( 1 - 79.8iT - 5.32e3T^{2} \)
79 \( 1 - 86.0T + 6.24e3T^{2} \)
83 \( 1 - 153.T + 6.88e3T^{2} \)
89 \( 1 - 83.3iT - 7.92e3T^{2} \)
97 \( 1 - 16.2iT - 9.40e3T^{2} \)
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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

−9.373110602068302195022531156267, −8.441887689288155337278535103649, −8.185047634089966236707963888381, −6.57431861114019517418373908951, −6.19085563996120485898148023177, −5.14953692699055981595364419028, −4.57335161816872088305168177042, −3.23022855125688218628020589911, −2.16619343926671446798710657730, −1.15915752580425530704392107021, 0.58660868887485405085622787513, 1.95302252812861833982847093996, 3.05820085706393558629189818387, 3.89475473374941977550584780040, 4.99188598398563413024890525111, 5.90861703749371010909741927882, 6.75255009728228556368351471518, 7.43575667569656862799606366972, 8.098831811552256489191249475511, 9.238508598941585718711575581747

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