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Properties

Label 2-164-164.147-c1-0-2
Degree $2$
Conductor $164$
Sign $-0.00799 - 0.999i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 + 0.799i)2-s + (0.663 + 1.60i)3-s + (0.722 − 1.86i)4-s + (1.22 + 0.624i)5-s + (−2.05 − 1.33i)6-s + (1.33 + 0.820i)7-s + (0.647 + 2.75i)8-s + (−0.00546 + 0.00546i)9-s + (−1.92 + 0.251i)10-s + (0.0312 + 0.396i)11-s + (3.46 − 0.0802i)12-s + (−1.80 − 0.434i)13-s + (−2.21 + 0.112i)14-s + (−0.187 + 2.37i)15-s + (−2.95 − 2.69i)16-s + (−3.70 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.824 + 0.565i)2-s + (0.383 + 0.925i)3-s + (0.361 − 0.932i)4-s + (0.548 + 0.279i)5-s + (−0.838 − 0.546i)6-s + (0.505 + 0.310i)7-s + (0.229 + 0.973i)8-s + (−0.00182 + 0.00182i)9-s + (−0.610 + 0.0793i)10-s + (0.00942 + 0.119i)11-s + (1.00 − 0.0231i)12-s + (−0.501 − 0.120i)13-s + (−0.592 + 0.0301i)14-s + (−0.0483 + 0.614i)15-s + (−0.739 − 0.673i)16-s + (−0.898 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00799 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00799 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.00799 - 0.999i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.00799 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692348 + 0.697903i\)
\(L(\frac12)\) \(\approx\) \(0.692348 + 0.697903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.799i)T \)
41 \( 1 + (-0.0405 + 6.40i)T \)
good3 \( 1 + (-0.663 - 1.60i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.22 - 0.624i)T + (2.93 + 4.04i)T^{2} \)
7 \( 1 + (-1.33 - 0.820i)T + (3.17 + 6.23i)T^{2} \)
11 \( 1 + (-0.0312 - 0.396i)T + (-10.8 + 1.72i)T^{2} \)
13 \( 1 + (1.80 + 0.434i)T + (11.5 + 5.90i)T^{2} \)
17 \( 1 + (3.70 - 4.33i)T + (-2.65 - 16.7i)T^{2} \)
19 \( 1 + (0.0591 + 0.246i)T + (-16.9 + 8.62i)T^{2} \)
23 \( 1 + (-0.527 - 0.383i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.35 - 1.58i)T + (-4.53 + 28.6i)T^{2} \)
31 \( 1 + (-3.22 + 9.93i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.115 + 0.355i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (2.48 - 0.394i)T + (40.8 - 13.2i)T^{2} \)
47 \( 1 + (-2.26 + 1.38i)T + (21.3 - 41.8i)T^{2} \)
53 \( 1 + (-9.34 + 7.98i)T + (8.29 - 52.3i)T^{2} \)
59 \( 1 + (6.80 - 9.36i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.04 - 0.798i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (-0.540 - 0.0425i)T + (66.1 + 10.4i)T^{2} \)
71 \( 1 + (10.9 - 0.862i)T + (70.1 - 11.1i)T^{2} \)
73 \( 1 + (-3.17 - 3.17i)T + 73iT^{2} \)
79 \( 1 + (0.650 - 0.269i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 + (2.54 - 4.15i)T + (-40.4 - 79.2i)T^{2} \)
97 \( 1 + (18.1 + 1.42i)T + (95.8 + 15.1i)T^{2} \)
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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

−13.36402429529870954300024246634, −11.79016991354026834427727752001, −10.59144816535915504074138265146, −9.984092304980700995293040444168, −9.061650363519449514380463306077, −8.199223880072748442949978301089, −6.83577918928333297074661433676, −5.66492479000638123057129527895, −4.33948088039289165741202666753, −2.24014672915357722912013897419, 1.42383563417754264504331486951, 2.68512642539412278557489150838, 4.70889663024024295437492711747, 6.68974432847453908641398568967, 7.52923983588965397409096793199, 8.504338064599219952131921119998, 9.470382156672864757955838064204, 10.54321854612425224401102455892, 11.61376625520122266919082453012, 12.53982064464708490563635403528

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