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Properties

Label 2-147-21.20-c5-0-56
Degree $2$
Conductor $147$
Sign $0.0853 + 0.996i$
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  − 2.13i·2-s + (9.16 − 12.6i)3-s + 27.4·4-s + 95.0·5-s + (−26.8 − 19.5i)6-s − 126. i·8-s + (−75.1 − 231. i)9-s − 202. i·10-s − 130. i·11-s + (251. − 346. i)12-s + 14.6i·13-s + (870. − 1.19e3i)15-s + 608.·16-s − 640.·17-s + (−492. + 160. i)18-s + 542. i·19-s + ⋯
L(s)  = 1  − 0.377i·2-s + (0.587 − 0.809i)3-s + 0.857·4-s + 1.70·5-s + (−0.305 − 0.221i)6-s − 0.700i·8-s + (−0.309 − 0.951i)9-s − 0.641i·10-s − 0.325i·11-s + (0.504 − 0.694i)12-s + 0.0240i·13-s + (0.999 − 1.37i)15-s + 0.593·16-s − 0.537·17-s + (−0.358 + 0.116i)18-s + 0.344i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0853 + 0.996i$
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: \((2,\ 147,\ (\ :5/2),\ 0.0853 + 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.849544106\)
\(L(\frac12)\) \(\approx\) \(3.849544106\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.16 + 12.6i)T \)
7 \( 1 \)
good2 \( 1 + 2.13iT - 32T^{2} \)
5 \( 1 - 95.0T + 3.12e3T^{2} \)
11 \( 1 + 130. iT - 1.61e5T^{2} \)
13 \( 1 - 14.6iT - 3.71e5T^{2} \)
17 \( 1 + 640.T + 1.41e6T^{2} \)
19 \( 1 - 542. iT - 2.47e6T^{2} \)
23 \( 1 - 4.09e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.59e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.90e3iT - 2.86e7T^{2} \)
37 \( 1 + 4.34e3T + 6.93e7T^{2} \)
41 \( 1 + 1.09e4T + 1.15e8T^{2} \)
43 \( 1 - 2.05e4T + 1.47e8T^{2} \)
47 \( 1 + 8.81e3T + 2.29e8T^{2} \)
53 \( 1 + 6.97e3iT - 4.18e8T^{2} \)
59 \( 1 + 4.00e4T + 7.14e8T^{2} \)
61 \( 1 - 3.32e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.43e3T + 1.35e9T^{2} \)
71 \( 1 + 3.24e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.27e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.59e3T + 3.07e9T^{2} \)
83 \( 1 + 3.16e4T + 3.93e9T^{2} \)
89 \( 1 - 3.19e4T + 5.58e9T^{2} \)
97 \( 1 + 1.43e5iT - 8.58e9T^{2} \)
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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

−12.02685534761396982416971007266, −10.88973816796091555512112991217, −9.813325779680108923245154315275, −8.968668835206162822021441425371, −7.49925337301640969472252210105, −6.46733347614874819006318333350, −5.66885186023694056588233084742, −3.25574839045158875377956196814, −2.12013705425074645662111200101, −1.33805201463579848369915760264, 1.93367724374874536721865426325, 2.77029800908917887853001853818, 4.75762443890300064790088600126, 5.88604551316514162632321585290, 6.86408238223668751090173142184, 8.350501789083639017193454807375, 9.361555119394636407084356192654, 10.29349460669293111931469906887, 10.96544383357300409481595135077, 12.53680689740479765286735248894

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