Properties

Label 2-147-21.20-c5-0-52
Degree $2$
Conductor $147$
Sign $-0.654 + 0.755i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.5i·3-s + 32·4-s − 243·9-s − 498. i·12-s − 1.14e3i·13-s + 1.02e3·16-s − 161. i·19-s − 3.12e3·25-s + 3.78e3i·27-s − 1.03e4i·31-s − 7.77e3·36-s + 6.66e3·37-s − 1.77e4·39-s − 2.24e4·43-s − 1.59e4i·48-s + ⋯
L(s)  = 1  − 0.999i·3-s + 4-s − 9-s − 0.999i·12-s − 1.87i·13-s + 16-s − 0.102i·19-s − 25-s + 1.00i·27-s − 1.93i·31-s − 36-s + 0.799·37-s − 1.87·39-s − 1.85·43-s − 0.999i·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.946943969\)
\(L(\frac12)\) \(\approx\) \(1.946943969\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5iT \)
7 \( 1 \)
good2 \( 1 - 32T^{2} \)
5 \( 1 + 3.12e3T^{2} \)
11 \( 1 - 1.61e5T^{2} \)
13 \( 1 + 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.41e6T^{2} \)
19 \( 1 + 161. iT - 2.47e6T^{2} \)
23 \( 1 - 6.43e6T^{2} \)
29 \( 1 - 2.05e7T^{2} \)
31 \( 1 + 1.03e4iT - 2.86e7T^{2} \)
37 \( 1 - 6.66e3T + 6.93e7T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 + 2.24e4T + 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 4.34e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.79e4T + 1.35e9T^{2} \)
71 \( 1 - 1.80e9T^{2} \)
73 \( 1 - 4.67e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.08e4T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 - 1.27e5iT - 8.58e9T^{2} \)
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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

−11.80383264807515841549356492128, −11.00677069816009224198216718914, −9.871762248068828598134417346377, −8.128821895189613885053504448728, −7.61818635230342723819810588003, −6.36362963533110108365532922463, −5.53488486678782408201045875137, −3.24023912076418603559457399214, −2.10094004331203923642477091921, −0.61319383602466123003363729961, 1.86243862887864202761484032072, 3.33781404314844089602731556852, 4.61818483850676143900332679746, 6.02292028923262869382215871363, 7.05305830452699616447645318448, 8.487735949416690918645448220058, 9.568641317143513811998128737777, 10.52176348900673877468694182112, 11.49734065026930261320698469912, 12.05209667589065974776508992130

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