↑

Properties

Label 2-164-164.163-c4-0-43
Degree $2$
Conductor $164$
Sign $0.859 - 0.511i$
Analytic cond. $16.9526$
Root an. cond. $4.11736$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.71 + 1.48i)2-s + 10.9·3-s + (11.5 − 11.0i)4-s + 8.39·5-s + (−40.5 + 16.2i)6-s + 64.7·7-s + (−26.4 + 58.2i)8-s + 38.0·9-s + (−31.1 + 12.5i)10-s + 53.4·11-s + (126. − 120. i)12-s + 211. i·13-s + (−240. + 96.3i)14-s + 91.6·15-s + (11.5 − 255. i)16-s + 86.3i·17-s + ⋯
L(s)  = 1  + (−0.928 + 0.372i)2-s + 1.21·3-s + (0.722 − 0.690i)4-s + 0.335·5-s + (−1.12 + 0.451i)6-s + 1.32·7-s + (−0.413 + 0.910i)8-s + 0.469·9-s + (−0.311 + 0.125i)10-s + 0.442·11-s + (0.876 − 0.837i)12-s + 1.25i·13-s + (−1.22 + 0.491i)14-s + 0.407·15-s + (0.0451 − 0.998i)16-s + 0.298i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(16.9526\)
Root analytic conductor: \(4.11736\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :2),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.265924953\)
\(L(\frac12)\) \(\approx\) \(2.265924953\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.71 - 1.48i)T \)
41 \( 1 + (-1.63e3 - 376. i)T \)
good3 \( 1 - 10.9T + 81T^{2} \)
5 \( 1 - 8.39T + 625T^{2} \)
7 \( 1 - 64.7T + 2.40e3T^{2} \)
11 \( 1 - 53.4T + 1.46e4T^{2} \)
13 \( 1 - 211. iT - 2.85e4T^{2} \)
17 \( 1 - 86.3iT - 8.35e4T^{2} \)
19 \( 1 - 370.T + 1.30e5T^{2} \)
23 \( 1 + 759. iT - 2.79e5T^{2} \)
29 \( 1 + 1.22e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.54e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.77e3T + 1.87e6T^{2} \)
43 \( 1 - 966. iT - 3.41e6T^{2} \)
47 \( 1 - 1.37e3T + 4.87e6T^{2} \)
53 \( 1 - 49.7iT - 7.89e6T^{2} \)
59 \( 1 - 2.18e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.86e3T + 1.38e7T^{2} \)
67 \( 1 + 1.14e3T + 2.01e7T^{2} \)
71 \( 1 + 4.66e3T + 2.54e7T^{2} \)
73 \( 1 - 8.28e3T + 2.83e7T^{2} \)
79 \( 1 + 5.69e3T + 3.89e7T^{2} \)
83 \( 1 + 3.04e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.30e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.20e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.92749215670756460965059576894, −11.12273982851756239434209397046, −9.817707649920329292797739980705, −9.006184048872003287151916409482, −8.246049873222663160484690756757, −7.41412685797953673335207988595, −6.06208423318766798723540914291, −4.44104076137228209994749116007, −2.47638942679101264518844946256, −1.44372944200937083116733159427, 1.24147439429115246295179011236, 2.46210055906629137413594689058, 3.68067685862193241813369920441, 5.59607829193167562394321332510, 7.61902102049159053378392679234, 7.87679357428340671907752303299, 9.074653211097575753956841181029, 9.723898660288218776578985018720, 11.01278704797115107138624635476, 11.78036769334921225248605733454

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