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Properties

Label 2-164-4.3-c2-0-37
Degree $2$
Conductor $164$
Sign $-0.926 - 0.377i$
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  + (0.384 − 1.96i)2-s − 4.91i·3-s + (−3.70 − 1.50i)4-s + 1.34·5-s + (−9.64 − 1.88i)6-s − 2.68i·7-s + (−4.38 + 6.68i)8-s − 15.1·9-s + (0.518 − 2.64i)10-s − 4.05i·11-s + (−7.41 + 18.1i)12-s + 18.6·13-s + (−5.26 − 1.03i)14-s − 6.62i·15-s + (11.4 + 11.1i)16-s − 4.56·17-s + ⋯
L(s)  = 1  + (0.192 − 0.981i)2-s − 1.63i·3-s + (−0.926 − 0.377i)4-s + 0.269·5-s + (−1.60 − 0.314i)6-s − 0.382i·7-s + (−0.548 + 0.836i)8-s − 1.68·9-s + (0.0518 − 0.264i)10-s − 0.368i·11-s + (−0.618 + 1.51i)12-s + 1.43·13-s + (−0.375 − 0.0736i)14-s − 0.441i·15-s + (0.715 + 0.699i)16-s − 0.268·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.926 - 0.377i$
Analytic conductor: \(4.46867\)
Root analytic conductor: \(2.11392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1),\ -0.926 - 0.377i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.268032 + 1.36770i\)
\(L(\frac12)\) \(\approx\) \(0.268032 + 1.36770i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.384 + 1.96i)T \)
41 \( 1 - 6.40T \)
good3 \( 1 + 4.91iT - 9T^{2} \)
5 \( 1 - 1.34T + 25T^{2} \)
7 \( 1 + 2.68iT - 49T^{2} \)
11 \( 1 + 4.05iT - 121T^{2} \)
13 \( 1 - 18.6T + 169T^{2} \)
17 \( 1 + 4.56T + 289T^{2} \)
19 \( 1 - 3.17iT - 361T^{2} \)
23 \( 1 - 5.32iT - 529T^{2} \)
29 \( 1 - 37.2T + 841T^{2} \)
31 \( 1 + 30.8iT - 961T^{2} \)
37 \( 1 + 57.0T + 1.36e3T^{2} \)
43 \( 1 + 50.7iT - 1.84e3T^{2} \)
47 \( 1 - 1.85iT - 2.20e3T^{2} \)
53 \( 1 + 33.3T + 2.80e3T^{2} \)
59 \( 1 + 44.9iT - 3.48e3T^{2} \)
61 \( 1 - 97.4T + 3.72e3T^{2} \)
67 \( 1 - 84.8iT - 4.48e3T^{2} \)
71 \( 1 - 81.7iT - 5.04e3T^{2} \)
73 \( 1 - 88.4T + 5.32e3T^{2} \)
79 \( 1 + 136. iT - 6.24e3T^{2} \)
83 \( 1 - 62.8iT - 6.88e3T^{2} \)
89 \( 1 - 151.T + 7.92e3T^{2} \)
97 \( 1 - 20.4T + 9.40e3T^{2} \)
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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

−12.08738849251372916164651293522, −11.34375144122440309130142203266, −10.31316098268235300816437659730, −8.843909807633128947376604700050, −8.021375118265258801143483867874, −6.59694070889406665704295476410, −5.63263505670847698047017126295, −3.70247066744958246836709531966, −2.12169849963723692263309248992, −0.877534080091758159060365998073, 3.40334054681764006488869668122, 4.49559031361976480414215512730, 5.48736483589269932805630420118, 6.54442051270845792542207699991, 8.316337512375829985972333005005, 9.022560399410292142516369863312, 9.942936383548633232607128759101, 10.89514833645472212160593556429, 12.22915734859360951343188891310, 13.57519419844470310568694742508

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