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Properties

Label 2-147-21.2-c2-0-11
Degree $2$
Conductor $147$
Sign $0.967 + 0.253i$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 − 0.653i)2-s + (2.97 + 0.387i)3-s + (−1.14 − 1.98i)4-s + (6.39 + 3.68i)5-s + (−3.11 − 2.38i)6-s + 8.22i·8-s + (8.69 + 2.30i)9-s + (−4.82 − 8.35i)10-s + (2.26 − 1.30i)11-s + (−2.63 − 6.34i)12-s − 6.35·13-s + (17.5 + 13.4i)15-s + (0.791 − 1.37i)16-s + (10.5 − 6.07i)17-s + (−8.34 − 8.29i)18-s + (5.11 − 8.85i)19-s + ⋯
L(s)  = 1  + (−0.565 − 0.326i)2-s + (0.991 + 0.129i)3-s + (−0.286 − 0.496i)4-s + (1.27 + 0.737i)5-s + (−0.519 − 0.397i)6-s + 1.02i·8-s + (0.966 + 0.256i)9-s + (−0.482 − 0.835i)10-s + (0.205 − 0.118i)11-s + (−0.219 − 0.528i)12-s − 0.488·13-s + (1.17 + 0.896i)15-s + (0.0494 − 0.0856i)16-s + (0.618 − 0.357i)17-s + (−0.463 − 0.460i)18-s + (0.269 − 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65030 - 0.212584i\)
\(L(\frac12)\) \(\approx\) \(1.65030 - 0.212584i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.97 - 0.387i)T \)
7 \( 1 \)
good2 \( 1 + (1.13 + 0.653i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-6.39 - 3.68i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.26 + 1.30i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 6.35T + 169T^{2} \)
17 \( 1 + (-10.5 + 6.07i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.11 + 8.85i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.72 - 2.15i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 17.3iT - 841T^{2} \)
31 \( 1 + (19.6 + 34.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (20.5 - 35.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8T + 1.84e3T^{2} \)
47 \( 1 + (34.6 + 19.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (90.9 - 52.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (35.8 - 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-10.2 + 17.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-13.5 - 23.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 + (30.3 + 52.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-31.6 + 54.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 89.9iT - 6.88e3T^{2} \)
89 \( 1 + (-54.7 - 31.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 19.1T + 9.40e3T^{2} \)
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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

−13.17542882993958291006530962009, −11.46453321512144493174207898221, −10.23311495945647725574734802663, −9.770970081302483534431345082038, −9.014699030087568649034479277952, −7.71258727835228309240320658667, −6.32100819890591233671128866332, −4.96711298127174081270606673787, −2.94406382814299143438036550315, −1.74173690841504507971087391381, 1.62637018303187796635495886606, 3.44565253404633293833937812612, 5.05426592136414247978894540931, 6.69289002659409301280337276239, 7.84188827687045326260647102849, 8.812416766405487716610813625931, 9.467087620735429931580615036414, 10.22573336612514589440438213577, 12.41203500206747818501919784271, 12.84039353663462928454455623210

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