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Properties

Label 2-164-164.111-c1-0-1
Degree $2$
Conductor $164$
Sign $-0.939 + 0.342i$
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  + (−0.221 + 1.39i)2-s + (−1.90 − 0.618i)4-s + (−3.93 + 2.00i)5-s + (1.28 − 2.52i)8-s + (−2.12 − 2.12i)9-s + (−1.92 − 5.93i)10-s + (1.04 + 4.33i)13-s + (3.23 + 2.35i)16-s + (−6.27 + 5.35i)17-s + (3.43 − 2.49i)18-s + (8.71 − 1.38i)20-s + (8.50 − 11.7i)25-s + (−6.28 + 0.494i)26-s + (5.72 + 4.88i)29-s + (−4 + 4i)32-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−1.75 + 0.895i)5-s + (0.453 − 0.891i)8-s + (−0.707 − 0.707i)9-s + (−0.609 − 1.87i)10-s + (0.288 + 1.20i)13-s + (0.809 + 0.587i)16-s + (−1.52 + 1.29i)17-s + (0.809 − 0.587i)18-s + (1.94 − 0.308i)20-s + (1.70 − 2.34i)25-s + (−1.23 + 0.0970i)26-s + (1.06 + 0.907i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0633156 - 0.358530i\)
\(L(\frac12)\) \(\approx\) \(0.0633156 - 0.358530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 - 1.39i)T \)
41 \( 1 + (6.39 + 0.297i)T \)
good3 \( 1 + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.93 - 2.00i)T + (2.93 - 4.04i)T^{2} \)
7 \( 1 + (-3.17 + 6.23i)T^{2} \)
11 \( 1 + (10.8 + 1.72i)T^{2} \)
13 \( 1 + (-1.04 - 4.33i)T + (-11.5 + 5.90i)T^{2} \)
17 \( 1 + (6.27 - 5.35i)T + (2.65 - 16.7i)T^{2} \)
19 \( 1 + (16.9 + 8.62i)T^{2} \)
23 \( 1 + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-5.72 - 4.88i)T + (4.53 + 28.6i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.23 - 6.87i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (40.8 + 13.2i)T^{2} \)
47 \( 1 + (-21.3 - 41.8i)T^{2} \)
53 \( 1 + (-2.53 + 2.96i)T + (-8.29 - 52.3i)T^{2} \)
59 \( 1 + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.07 - 0.962i)T + (58.0 - 18.8i)T^{2} \)
67 \( 1 + (-66.1 + 10.4i)T^{2} \)
71 \( 1 + (-70.1 - 11.1i)T^{2} \)
73 \( 1 + (0.538 - 0.538i)T - 73iT^{2} \)
79 \( 1 + (-55.8 - 55.8i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (2.65 - 1.62i)T + (40.4 - 79.2i)T^{2} \)
97 \( 1 + (-0.439 - 5.58i)T + (-95.8 + 15.1i)T^{2} \)
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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

−13.71587610208514878201719934880, −12.28714370669162747937898572410, −11.41538310797861849622826540096, −10.46761560052722854314728518960, −8.784269035080205796500801546516, −8.319463634678402802743884907957, −6.89715969552752269899718670369, −6.48073644870006459075885606485, −4.48245395875141134538686471084, −3.54542777908827734919233828216, 0.36324009055790117107441458093, 2.90753151732982403958588325650, 4.28606487650745215684607153785, 5.17135252001415058671581780640, 7.54312094962184118508489956597, 8.349532829359873826196879038085, 9.039244727456703176893924255886, 10.65701004362550435989886743819, 11.38526596747991724990534682263, 12.07757922073379447545086701594

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