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Properties

Label 2-147-7.4-c5-0-28
Degree $2$
Conductor $147$
Sign $-0.991 + 0.126i$
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  + (0.395 + 0.684i)2-s + (4.5 − 7.79i)3-s + (15.6 − 27.1i)4-s + (−52.0 − 90.2i)5-s + 7.11·6-s + 50.1·8-s + (−40.5 − 70.1i)9-s + (41.1 − 71.3i)10-s + (248. − 430. i)11-s + (−141. − 244. i)12-s + 206.·13-s − 937.·15-s + (−482. − 835. i)16-s + (31.5 − 54.6i)17-s + (32.0 − 55.4i)18-s + (661. + 1.14e3i)19-s + ⋯
L(s)  = 1  + (0.0698 + 0.121i)2-s + (0.288 − 0.499i)3-s + (0.490 − 0.849i)4-s + (−0.931 − 1.61i)5-s + 0.0807·6-s + 0.276·8-s + (−0.166 − 0.288i)9-s + (0.130 − 0.225i)10-s + (0.620 − 1.07i)11-s + (−0.283 − 0.490i)12-s + 0.338·13-s − 1.07·15-s + (−0.470 − 0.815i)16-s + (0.0265 − 0.0459i)17-s + (0.0232 − 0.0403i)18-s + (0.420 + 0.728i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.867406822\)
\(L(\frac12)\) \(\approx\) \(1.867406822\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-0.395 - 0.684i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (52.0 + 90.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-248. + 430. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 206.T + 3.71e5T^{2} \)
17 \( 1 + (-31.5 + 54.6i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-661. - 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-97.2 - 168. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + (3.76e3 - 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 4.18e3T + 1.15e8T^{2} \)
43 \( 1 - 5.96e3T + 1.47e8T^{2} \)
47 \( 1 + (2.19e3 + 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.89e3 - 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.75e3 + 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (5.31e3 + 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-6.63e3 + 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.88e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.56e4 + 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.97e4 + 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + (5.64e4 + 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 3.03e4T + 8.58e9T^{2} \)
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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

−11.83343391490364981000777639055, −10.84726101585602912697721895784, −9.271989989653898454165823910815, −8.529685605599663516265393905571, −7.48700538365522744423094121686, −6.10877602407146541108154319102, −5.05307791406769457207849168895, −3.61824299976399470164633089295, −1.46994270815991637902921007129, −0.62445258124539311716451783425, 2.41655969477833661649222774965, 3.42385713808556408189656861097, 4.31752645131620556006845878574, 6.58840893187143643147768595301, 7.29185791633888445297380321127, 8.233168868560332888258640586218, 9.716120420431017603440657761599, 10.85194713320570252959714424415, 11.47909389830708197241721965089, 12.34338230244581497594595244782

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