Properties

Label 2-147-7.2-c3-0-11
Degree $2$
Conductor $147$
Sign $0.991 + 0.126i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)2-s + (1.5 + 2.59i)3-s + (−0.5 − 0.866i)4-s + (−1.5 + 2.59i)5-s + 9·6-s + 21·8-s + (−4.5 + 7.79i)9-s + (4.5 + 7.79i)10-s + (7.5 + 12.9i)11-s + (1.50 − 2.59i)12-s + 64·13-s − 9·15-s + (35.5 − 61.4i)16-s + (42 + 72.7i)17-s + (13.5 + 23.3i)18-s + (−8 + 13.8i)19-s + ⋯
L(s)  = 1  + (0.530 − 0.918i)2-s + (0.288 + 0.499i)3-s + (−0.0625 − 0.108i)4-s + (−0.134 + 0.232i)5-s + 0.612·6-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (0.142 + 0.246i)10-s + (0.205 + 0.356i)11-s + (0.0360 − 0.0625i)12-s + 1.36·13-s − 0.154·15-s + (0.554 − 0.960i)16-s + (0.599 + 1.03i)17-s + (0.176 + 0.306i)18-s + (−0.0965 + 0.167i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.65384 - 0.168413i\)
\(L(\frac12)\) \(\approx\) \(2.65384 - 0.168413i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (-1.5 + 2.59i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 64T + 2.19e3T^{2} \)
17 \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (8 - 13.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 297T + 2.43e4T^{2} \)
31 \( 1 + (126.5 + 219. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-158 + 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 360T + 6.89e4T^{2} \)
43 \( 1 - 26T + 7.95e4T^{2} \)
47 \( 1 + (15 - 25.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (181.5 + 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (59 - 102. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-185 - 320. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 342T + 3.57e5T^{2} \)
73 \( 1 + (-181 - 313. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (233.5 - 404. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 477T + 5.71e5T^{2} \)
89 \( 1 + (-453 + 784. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 503T + 9.12e5T^{2} \)
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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

−12.70896822669867857549390891060, −11.31591211882210635298441118862, −10.89723522390686879738388645702, −9.764356135350725708438517771339, −8.505395791247484752971236149756, −7.33475489140444438216219836936, −5.70546428097437895329684621541, −4.10924363295733866121086820539, −3.40932366894813089321204557860, −1.78412743762629322276598970783, 1.32128003035094388887531434919, 3.49140915880193765976326389736, 5.06508962445991743858441987979, 6.14557377251303475641347644439, 7.12422991627852628487606547170, 8.131431321827146537120156053627, 9.210009467697763846336108466812, 10.72163773940034770007008903459, 11.71474196055688402396418224015, 13.05253585052028495242296659509

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