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Properties

Label 2-147-21.20-c5-0-41
Degree $2$
Conductor $147$
Sign $0.999 + 0.0191i$
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  + 9.64i·2-s + (−2.14 − 15.4i)3-s − 60.9·4-s + 95.0·5-s + (148. − 20.6i)6-s − 279. i·8-s + (−233. + 66.2i)9-s + 916. i·10-s − 151. i·11-s + (130. + 941. i)12-s − 901. i·13-s + (−203. − 1.46e3i)15-s + 741.·16-s − 488.·17-s + (−638. − 2.25e3i)18-s − 2.18e3i·19-s + ⋯
L(s)  = 1  + 1.70i·2-s + (−0.137 − 0.990i)3-s − 1.90·4-s + 1.69·5-s + (1.68 − 0.234i)6-s − 1.54i·8-s + (−0.962 + 0.272i)9-s + 2.89i·10-s − 0.378i·11-s + (0.262 + 1.88i)12-s − 1.47i·13-s + (−0.233 − 1.68i)15-s + 0.723·16-s − 0.410·17-s + (−0.464 − 1.63i)18-s − 1.38i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.999 + 0.0191i$
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: \((2,\ 147,\ (\ :5/2),\ 0.999 + 0.0191i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.825632935\)
\(L(\frac12)\) \(\approx\) \(1.825632935\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.14 + 15.4i)T \)
7 \( 1 \)
good2 \( 1 - 9.64iT - 32T^{2} \)
5 \( 1 - 95.0T + 3.12e3T^{2} \)
11 \( 1 + 151. iT - 1.61e5T^{2} \)
13 \( 1 + 901. iT - 3.71e5T^{2} \)
17 \( 1 + 488.T + 1.41e6T^{2} \)
19 \( 1 + 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.38e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.15e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.25e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.68e3T + 6.93e7T^{2} \)
41 \( 1 - 1.44e4T + 1.15e8T^{2} \)
43 \( 1 - 1.00e4T + 1.47e8T^{2} \)
47 \( 1 + 9.27e3T + 2.29e8T^{2} \)
53 \( 1 + 1.67e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.36e3T + 7.14e8T^{2} \)
61 \( 1 + 7.94e3iT - 8.44e8T^{2} \)
67 \( 1 - 3.39e4T + 1.35e9T^{2} \)
71 \( 1 - 4.31e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.15e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.01e4T + 3.07e9T^{2} \)
83 \( 1 + 2.96e3T + 3.93e9T^{2} \)
89 \( 1 - 8.44e4T + 5.58e9T^{2} \)
97 \( 1 - 6.07e4iT - 8.58e9T^{2} \)
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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

−12.89796398173335622067986303352, −11.05197905382941303421382521052, −9.642864148641166433128842950506, −8.688955874199881131915097068579, −7.66261273740595979587319962994, −6.59228798170076635619414311064, −5.85190543335474357964813628428, −5.17448688490666340773173321545, −2.49336472146307055046944572943, −0.64011741316692873716553725553, 1.53322243838568866608213818113, 2.52597464959104536565048291239, 3.99357363976761076946794151607, 5.06377137379445723849946700962, 6.34382048305215875781119362530, 8.925699160112381063933803323444, 9.410904107533237671394405150150, 10.24427836052020621560809087752, 10.85400095125928422651274356805, 11.99783767527129710518877123308

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