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Properties

Label 2-164-41.14-c2-0-1
Degree $2$
Conductor $164$
Sign $-0.804 - 0.593i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 3.32i)3-s + (−4.86 + 4.86i)5-s + (−0.491 − 1.18i)7-s + (−2.81 + 2.81i)9-s + (−11.9 + 4.94i)11-s + (1.14 + 2.75i)13-s + (−22.8 − 9.47i)15-s + (−0.425 + 1.02i)17-s + (4.11 − 9.92i)19-s + (3.27 − 3.27i)21-s + 27.0i·23-s − 22.2i·25-s + (16.6 + 6.91i)27-s + (7.91 + 19.1i)29-s + 29.4i·31-s + ⋯
L(s)  = 1  + (0.459 + 1.10i)3-s + (−0.972 + 0.972i)5-s + (−0.0702 − 0.169i)7-s + (−0.313 + 0.313i)9-s + (−1.08 + 0.449i)11-s + (0.0878 + 0.212i)13-s + (−1.52 − 0.631i)15-s + (−0.0250 + 0.0603i)17-s + (0.216 − 0.522i)19-s + (0.155 − 0.155i)21-s + 1.17i·23-s − 0.889i·25-s + (0.618 + 0.256i)27-s + (0.272 + 0.658i)29-s + 0.949i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(4.46867\)
Root analytic conductor: \(2.11392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1),\ -0.804 - 0.593i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.359648 + 1.09277i\)
\(L(\frac12)\) \(\approx\) \(0.359648 + 1.09277i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (31.0 + 26.7i)T \)
good3 \( 1 + (-1.37 - 3.32i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (4.86 - 4.86i)T - 25iT^{2} \)
7 \( 1 + (0.491 + 1.18i)T + (-34.6 + 34.6i)T^{2} \)
11 \( 1 + (11.9 - 4.94i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.14 - 2.75i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + (0.425 - 1.02i)T + (-204. - 204. i)T^{2} \)
19 \( 1 + (-4.11 + 9.92i)T + (-255. - 255. i)T^{2} \)
23 \( 1 - 27.0iT - 529T^{2} \)
29 \( 1 + (-7.91 - 19.1i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 29.4iT - 961T^{2} \)
37 \( 1 - 23.4T + 1.36e3T^{2} \)
43 \( 1 + (-43.3 - 43.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (9.55 - 23.0i)T + (-1.56e3 - 1.56e3i)T^{2} \)
53 \( 1 + (-32.2 + 13.3i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 - 114.T + 3.48e3T^{2} \)
61 \( 1 + (35.1 + 35.1i)T + 3.72e3iT^{2} \)
67 \( 1 + (-16.2 + 39.1i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (0.0637 + 0.153i)T + (-3.56e3 + 3.56e3i)T^{2} \)
73 \( 1 + (74.9 + 74.9i)T + 5.32e3iT^{2} \)
79 \( 1 + (48.4 - 20.0i)T + (4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 - 22.6T + 6.88e3T^{2} \)
89 \( 1 + (35.9 + 86.8i)T + (-5.60e3 + 5.60e3i)T^{2} \)
97 \( 1 + (-48.3 - 20.0i)T + (6.65e3 + 6.65e3i)T^{2} \)
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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

−13.06579125353850513321532983398, −11.76200182132955747409989514879, −10.79329488104081700827117763894, −10.15231733709450701441466787277, −9.049305166283753185913774111141, −7.79823219821276543248337739337, −6.91803742539714979035736745877, −5.07600178024257821265030774302, −3.87121703490541226179814039164, −2.92700108795208005944622496276, 0.68293168543013450494812099459, 2.56974000148990396700379237562, 4.30176002685890108865037335715, 5.72996370905263250305802875354, 7.24775581061368429397049947323, 8.136493170290235623477029967875, 8.593143607585678888927598982737, 10.20450278738807833205284294237, 11.54557298777549213441030250531, 12.44802524880867210866594014508

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