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Properties

Label 2-164-164.11-c1-0-11
Degree $2$
Conductor $164$
Sign $0.980 + 0.198i$
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  + (−1.23 + 0.686i)2-s + (−0.869 + 2.09i)3-s + (1.05 − 1.69i)4-s + (1.95 − 3.84i)5-s + (−0.365 − 3.19i)6-s + (−0.861 − 3.58i)7-s + (−0.143 + 2.82i)8-s + (−1.52 − 1.52i)9-s + (0.215 + 6.09i)10-s + (1.33 − 1.56i)11-s + (2.64 + 3.69i)12-s + (−0.539 − 0.880i)13-s + (3.52 + 3.84i)14-s + (6.35 + 7.44i)15-s + (−1.76 − 3.59i)16-s + (0.209 + 2.66i)17-s + ⋯
L(s)  = 1  + (−0.874 + 0.485i)2-s + (−0.501 + 1.21i)3-s + (0.529 − 0.848i)4-s + (0.875 − 1.71i)5-s + (−0.149 − 1.30i)6-s + (−0.325 − 1.35i)7-s + (−0.0508 + 0.998i)8-s + (−0.509 − 0.509i)9-s + (0.0681 + 1.92i)10-s + (0.403 − 0.472i)11-s + (0.762 + 1.06i)12-s + (−0.149 − 0.244i)13-s + (0.943 + 1.02i)14-s + (1.64 + 1.92i)15-s + (−0.440 − 0.897i)16-s + (0.0509 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.980 + 0.198i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ 0.980 + 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720975 - 0.0721207i\)
\(L(\frac12)\) \(\approx\) \(0.720975 - 0.0721207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.686i)T \)
41 \( 1 + (-6.09 - 1.95i)T \)
good3 \( 1 + (0.869 - 2.09i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.95 + 3.84i)T + (-2.93 - 4.04i)T^{2} \)
7 \( 1 + (0.861 + 3.58i)T + (-6.23 + 3.17i)T^{2} \)
11 \( 1 + (-1.33 + 1.56i)T + (-1.72 - 10.8i)T^{2} \)
13 \( 1 + (0.539 + 0.880i)T + (-5.90 + 11.5i)T^{2} \)
17 \( 1 + (-0.209 - 2.66i)T + (-16.7 + 2.65i)T^{2} \)
19 \( 1 + (0.0153 + 0.00939i)T + (8.62 + 16.9i)T^{2} \)
23 \( 1 + (-3.82 - 2.78i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.347 - 4.41i)T + (-28.6 - 4.53i)T^{2} \)
31 \( 1 + (-2.93 + 9.03i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.09 - 3.36i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.739 - 4.66i)T + (-40.8 + 13.2i)T^{2} \)
47 \( 1 + (0.161 - 0.671i)T + (-41.8 - 21.3i)T^{2} \)
53 \( 1 + (-0.621 - 0.0489i)T + (52.3 + 8.29i)T^{2} \)
59 \( 1 + (-1.67 + 2.31i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.0314 - 0.198i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (7.84 - 6.69i)T + (10.4 - 66.1i)T^{2} \)
71 \( 1 + (1.54 + 1.31i)T + (11.1 + 70.1i)T^{2} \)
73 \( 1 + (-3.34 + 3.34i)T - 73iT^{2} \)
79 \( 1 + (-1.95 - 0.808i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + (-3.06 + 0.735i)T + (79.2 - 40.4i)T^{2} \)
97 \( 1 + (2.16 - 1.84i)T + (15.1 - 95.8i)T^{2} \)
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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

−12.94420673454150693911396532136, −11.38984763904957931541158421949, −10.40710904659518631225767282348, −9.700972835010002916683201059550, −9.066379347467742541282431204720, −7.85762033650488489019074768937, −6.21620014686915163002223092115, −5.24166684925284605309450905018, −4.22848627941715884531275932651, −1.04448800036277838420688081802, 2.02213019037548403384226001028, 2.88123109113601490367190261003, 5.94014542704658664631420565566, 6.73074405475696362555500513924, 7.35834880186529173423562151026, 8.985507574673296810861074566316, 9.884184616940509132313911520387, 10.95779984970161072824766319072, 11.85587170769112525788394100499, 12.50524216810893568570127031513

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