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Properties

Label 1-147-147.65-r1-0-0
Degree $1$
Conductor $147$
Sign $-0.961 + 0.274i$
Analytic cond. $15.7973$
Root an. cond. $15.7973$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (0.365 + 0.930i)10-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.988 − 0.149i)16-s + (−0.826 + 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.623 − 0.781i)8-s + (0.365 + 0.930i)10-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.988 − 0.149i)16-s + (−0.826 + 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09771407465 - 0.6988839104i\)
\(L(\frac12)\) \(\approx\) \(-0.09771407465 - 0.6988839104i\)
\(L(1)\) \(\approx\) \(0.9224870084 - 0.4743442023i\)
\(L(1)\) \(\approx\) \(0.9224870084 - 0.4743442023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
97 \( 1 + T \)
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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

−28.58824072500259174085127439190, −27.30202918196829211504800401133, −26.35124304318213319984208136635, −25.34201294579777620312281840443, −24.33940971310540571541787605041, −23.68661114345443384062312914023, −22.82225674395362071359241361629, −21.51170201125282350421567974419, −20.8126246464996880316214895661, −19.75652493916790723047193097928, −18.25705759744783401907587080614, −17.11791431812722835074551608547, −16.17069062809287574002296932321, −15.517151463378528312483302987315, −14.21927419391293605666274288164, −13.19741049504745113314765892378, −12.33361948513602005088331355837, −11.34703465979114499169942344235, −9.518490580325445839349745362649, −8.31876426657521774655696444158, −7.42880767886656938211828188721, −6.03272722182290559201215559495, −4.83057942190442233651851091829, −4.01101362764519720049449027986, −2.229242628848605477192127137780, 0.192666614528067454616038348107, 2.324735105353978553366930406191, 3.24724983823265663132867724974, 4.59882982292419355818734294288, 5.91138050284395234779467838149, 7.06943150459538057468194183704, 8.56290325790506621740904436959, 10.43124813650426495229936268755, 10.645923298992077821147191198871, 11.99864595676947919489620226265, 13.050179549067002742394174527055, 14.04431836277752854465257726769, 15.194906849597995617941295519685, 15.74284401243159009690563822021, 17.711249199462100553573709372368, 18.58546389434196818313133932234, 19.59479039455782952018965655376, 20.40458167255031790007496237786, 21.767367484113678588814893354263, 22.230055155884823488456022836527, 23.42385543374043156938315778418, 24.004544122031524890282932150308, 25.42282534123802479461810322140, 26.56298377653443620610427964392, 27.50501492335061133654795613355

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