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Properties

Label 1-164-164.51-r1-0-0
Degree $1$
Conductor $164$
Sign $-0.789 - 0.613i$
Analytic cond. $17.6242$
Root an. cond. $17.6242$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + 9-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s − 27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  − 3-s + (−0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + 9-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s − 27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(17.6242\)
Root analytic conductor: \(17.6242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 164,\ (1:\ ),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06930118696 + 0.2022994634i\)
\(L(\frac12)\) \(\approx\) \(-0.06930118696 + 0.2022994634i\)
\(L(1)\) \(\approx\) \(0.5426099902 + 0.1955609918i\)
\(L(1)\) \(\approx\) \(0.5426099902 + 0.1955609918i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.154702957623909554869482386803, −26.133508501298166089490795262374, −24.52005059345105982940844726720, −23.968694739380704083994358904967, −22.984112355445459607963408221718, −22.31766768991007491888699055668, −21.058609038767451295074404431212, −19.85236611376168768329368971673, −19.23606657683098774029056450593, −17.63178359483562008032802484520, −17.047916665917467325872814436834, −16.04342454706864037252338926361, −15.2634787982290545425034506888, −13.475398356121451558470389610397, −12.76019747252671298526147826422, −11.47430993020050832222969072199, −10.89582918912496828299499194742, −9.54606739288160637281923228947, −8.13827858777053689212993081353, −6.99682522086192048957324156393, −5.89369162342463846054659224557, −4.503737297237219098435912785917, −3.659230163790265867784148224017, −1.13887113230158540520826932677, −0.10483228147495455562415376401, 1.93165108062744704876592939768, 3.73206955965324465076409400115, 4.844131044825073364661358570867, 6.38834074070325246808668596776, 6.90856447644482878445202764059, 8.54666437980800875861029551468, 9.80650110856760636197119238227, 11.05010836582952971257062224517, 11.90475077671951043679511261873, 12.50105988627631046976047449885, 14.21243313557077679694756390445, 15.349791743367860308259159621723, 16.098811556185286212142488898407, 17.1438159712171756488561655185, 18.49728931601436738515635955362, 18.80292753089000744877729169126, 20.17600921326482453467320079111, 21.5804225796325222963134079719, 22.43557794741702556762272752758, 22.95942674892455024613979951406, 24.08398848549482626663035670144, 25.00905581353001767494897768743, 26.295150383680386287844992877462, 27.28005242890698494378432630380, 28.07358603713821083262001028150

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