↑

Properties

Label 2-164-4.3-c2-0-27
Degree $2$
Conductor $164$
Sign $0.419 + 0.907i$
Analytic cond. $4.46867$
Root an. cond. $2.11392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 − 1.68i)2-s + 0.360i·3-s + (−1.67 − 3.63i)4-s + 8.57·5-s + (0.607 + 0.388i)6-s + 6.08i·7-s + (−7.92 − 1.08i)8-s + 8.86·9-s + (9.23 − 14.4i)10-s − 8.60i·11-s + (1.30 − 0.605i)12-s − 3.09·13-s + (10.2 + 6.56i)14-s + 3.09i·15-s + (−10.3 + 12.1i)16-s − 20.0·17-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)2-s + 0.120i·3-s + (−0.419 − 0.907i)4-s + 1.71·5-s + (0.101 + 0.0647i)6-s + 0.869i·7-s + (−0.990 − 0.135i)8-s + 0.985·9-s + (0.923 − 1.44i)10-s − 0.782i·11-s + (0.109 − 0.0504i)12-s − 0.238·13-s + (0.732 + 0.468i)14-s + 0.206i·15-s + (−0.647 + 0.761i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(4.46867\)
Root analytic conductor: \(2.11392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1),\ 0.419 + 0.907i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96470 - 1.25630i\)
\(L(\frac12)\) \(\approx\) \(1.96470 - 1.25630i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 + 1.68i)T \)
41 \( 1 + 6.40T \)
good3 \( 1 - 0.360iT - 9T^{2} \)
5 \( 1 - 8.57T + 25T^{2} \)
7 \( 1 - 6.08iT - 49T^{2} \)
11 \( 1 + 8.60iT - 121T^{2} \)
13 \( 1 + 3.09T + 169T^{2} \)
17 \( 1 + 20.0T + 289T^{2} \)
19 \( 1 + 27.0iT - 361T^{2} \)
23 \( 1 - 29.7iT - 529T^{2} \)
29 \( 1 - 4.38T + 841T^{2} \)
31 \( 1 + 2.89iT - 961T^{2} \)
37 \( 1 + 55.4T + 1.36e3T^{2} \)
43 \( 1 - 41.4iT - 1.84e3T^{2} \)
47 \( 1 - 68.1iT - 2.20e3T^{2} \)
53 \( 1 + 41.7T + 2.80e3T^{2} \)
59 \( 1 - 57.7iT - 3.48e3T^{2} \)
61 \( 1 - 33.6T + 3.72e3T^{2} \)
67 \( 1 + 31.9iT - 4.48e3T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + 27.9T + 5.32e3T^{2} \)
79 \( 1 - 58.7iT - 6.24e3T^{2} \)
83 \( 1 + 132. iT - 6.88e3T^{2} \)
89 \( 1 + 135.T + 7.92e3T^{2} \)
97 \( 1 - 2.64T + 9.40e3T^{2} \)
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

−12.65589592366033773065401561173, −11.37235262386156606034074692162, −10.47468680137157747532934464602, −9.422794480352075213569049016701, −9.003781996216649573781422045741, −6.67014695742645343920734557144, −5.69372386022664651188869991936, −4.74347586478607370243788362583, −2.83941543108273356503790472008, −1.68243124683375330293844304869, 2.02526794980847563141556844744, 4.12309025143126321767766552823, 5.23242387692795830634756176398, 6.55468672955292381937165643805, 7.08890538210465605067753880883, 8.590306245272520094396855156404, 9.836503297335594264473186996674, 10.42041466568551362608000704006, 12.36521677600899897287574086994, 13.01339111899379688140138192825

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