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Properties

Label 2-164-164.163-c4-0-6
Degree $2$
Conductor $164$
Sign $-0.903 - 0.428i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s + 14·5-s − 64·8-s − 81·9-s − 56·10-s + 240i·13-s + 256·16-s − 480i·17-s + 324·18-s + 224·20-s − 429·25-s − 960i·26-s + 1.68e3i·29-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 2-s + 4-s + 0.560·5-s − 8-s − 9-s − 0.560·10-s + 1.42i·13-s + 16-s − 1.66i·17-s + 18-s + 0.560·20-s − 0.686·25-s − 1.42i·26-s + 1.99i·29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.903 - 0.428i$
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: \((2,\ 164,\ (\ :2),\ -0.903 - 0.428i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2962013305\)
\(L(\frac12)\) \(\approx\) \(0.2962013305\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
41 \( 1 + (1.51e3 + 720i)T \)
good3 \( 1 + 81T^{2} \)
5 \( 1 - 14T + 625T^{2} \)
7 \( 1 + 2.40e3T^{2} \)
11 \( 1 + 1.46e4T^{2} \)
13 \( 1 - 240iT - 2.85e4T^{2} \)
17 \( 1 + 480iT - 8.35e4T^{2} \)
19 \( 1 + 1.30e5T^{2} \)
23 \( 1 - 2.79e5T^{2} \)
29 \( 1 - 1.68e3iT - 7.07e5T^{2} \)
31 \( 1 - 9.23e5T^{2} \)
37 \( 1 + 2.16e3T + 1.87e6T^{2} \)
43 \( 1 - 3.41e6T^{2} \)
47 \( 1 + 4.87e6T^{2} \)
53 \( 1 - 5.04e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.21e7T^{2} \)
61 \( 1 + 6.95e3T + 1.38e7T^{2} \)
67 \( 1 + 2.01e7T^{2} \)
71 \( 1 + 2.54e7T^{2} \)
73 \( 1 + 1.44e3T + 2.83e7T^{2} \)
79 \( 1 + 3.89e7T^{2} \)
83 \( 1 - 4.74e7T^{2} \)
89 \( 1 + 1.24e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.87e4iT - 8.85e7T^{2} \)
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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

−12.15000576741899867884905807256, −11.49245555791443824268279490407, −10.48655479206905890769496439481, −9.267339514387753967441047743248, −8.862619056673247085625394703473, −7.39180717915875479502852447171, −6.45938034824675917137142452350, −5.20217020016027327227471323874, −3.05725846453761676013542335522, −1.71563756104478521507851507905, 0.14448920915642909908543719449, 1.91142141695570175259475171900, 3.30137606874165206178683922708, 5.59951820810910098489085300036, 6.31812317164349689255038562760, 7.939425638701714590311501716604, 8.490318423099431330109187619813, 9.800954971588268725163091697289, 10.48889056071236418047090003162, 11.49051169917892777840272912019

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