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Properties

Label 1-147-147.32-r1-0-0
Degree $1$
Conductor $147$
Sign $0.687 + 0.725i$
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  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)10-s + (−0.365 + 0.930i)11-s + (0.623 + 0.781i)13-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)10-s + (−0.365 + 0.930i)11-s + (0.623 + 0.781i)13-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.687 + 0.725i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ 0.687 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.132836453 + 1.346964270i\)
\(L(\frac12)\) \(\approx\) \(3.132836453 + 1.346964270i\)
\(L(1)\) \(\approx\) \(1.991480409 + 0.3531710873i\)
\(L(1)\) \(\approx\) \(1.991480409 + 0.3531710873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (-0.0747 + 0.997i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
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

−28.09259915252157381410683910082, −26.72525054669095765890090294251, −25.469135652861040990639141300713, −24.73677642940831252567970711137, −23.741111017801745672747145245463, −23.14641649882718474607821292640, −21.7204931530459996288308085324, −21.07209638009614083779558607955, −20.13646696783012063778938704862, −19.11141117778766623798461479205, −17.446931030869087556662378360736, −16.449752623271880566654975111951, −15.69815951644537737490618861685, −14.56464991940217005482843774570, −13.22808870056538931630767181540, −12.825460300904016646213933488766, −11.51724560679102627448297350628, −10.49666072635610463285764960723, −8.70285011633074242704463766421, −7.82082306698328923976433623828, −6.193909926665471039276692891204, −5.30916833492034645425029574981, −4.12955423825746243748382460731, −2.837688309436154414463075975582, −1.01587915255743241871826978491, 1.7984453865789075885653670167, 3.081387346346899848220833906763, 4.21534623998505444742801965373, 5.61473508799935791681717268379, 6.78524547553867521674969313218, 7.64570106313385736098324200258, 9.668802377234727043668968909036, 10.74543526409218842124531969594, 11.65232117072979616665249413482, 12.753741135151931227269342123144, 13.944486745501657453815264673754, 14.719135391971568217285828615359, 15.622126793036922962536480993636, 16.76372887152911836648099586940, 18.32502713258473047125073121074, 19.108606543056641951551427331044, 20.39980425309498808172605642614, 21.23721521578480783567217823433, 22.2350968992494358060729438856, 23.2041343765531754632036519485, 23.63907060138878391420705131621, 25.32913984134117598185389757816, 25.67690565526350449850792592982, 27.08041460345535019050971893203, 28.270807976015338973968019870714

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