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Properties

Label 2-147-7.4-c5-0-10
Degree $2$
Conductor $147$
Sign $0.968 + 0.250i$
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  + (−2.5 − 4.33i)2-s + (4.5 − 7.79i)3-s + (3.49 − 6.06i)4-s + (47 + 81.4i)5-s − 45.0·6-s − 194.·8-s + (−40.5 − 70.1i)9-s + (235 − 407. i)10-s + (−26 + 45.0i)11-s + (−31.4 − 54.5i)12-s + 770·13-s + 846·15-s + (375.5 + 650. i)16-s + (−1.01e3 + 1.75e3i)17-s + (−202. + 350. i)18-s + (866 + 1.49e3i)19-s + ⋯
L(s)  = 1  + (−0.441 − 0.765i)2-s + (0.288 − 0.499i)3-s + (0.109 − 0.189i)4-s + (0.840 + 1.45i)5-s − 0.510·6-s − 1.07·8-s + (−0.166 − 0.288i)9-s + (0.743 − 1.28i)10-s + (−0.0647 + 0.112i)11-s + (−0.0631 − 0.109i)12-s + 1.26·13-s + 0.970·15-s + (0.366 + 0.635i)16-s + (−0.848 + 1.46i)17-s + (−0.147 + 0.255i)18-s + (0.550 + 0.953i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
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: \((2,\ 147,\ (\ :5/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.968581824\)
\(L(\frac12)\) \(\approx\) \(1.968581824\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.5 + 4.33i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-47 - 81.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (26 - 45.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 770T + 3.71e5T^{2} \)
17 \( 1 + (1.01e3 - 1.75e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-866 - 1.49e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-288 - 498. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.51e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.16e3 + 5.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.66e3 - 6.35e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 3.26e3T + 1.15e8T^{2} \)
43 \( 1 - 5.42e3T + 1.47e8T^{2} \)
47 \( 1 + (-432 - 748. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.09e3 - 3.62e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (5.61e3 - 9.71e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.28e4 + 3.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (698 - 1.20e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.87e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.31e4 + 4.01e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.87e4 + 8.43e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 8.12e4T + 3.93e9T^{2} \)
89 \( 1 + (1.59e3 + 2.75e3i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 4.91e3T + 8.58e9T^{2} \)
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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

−11.83475254242999799266085534577, −10.82820201141634930743419504308, −10.32782075288812437657158866148, −9.267387346650376396977058175321, −8.013309947817352150898917328868, −6.35651360040168088867226809322, −6.15467421066260826311314829097, −3.48636931302951826633164570845, −2.37115858524995670670140819514, −1.37177542624707905147014797211, 0.808257275134891997117419598196, 2.75542194670253572542586028603, 4.56467682597670960062182026942, 5.63842212043762789204939678917, 6.85417060018381406090809122180, 8.340378550504272466259569355962, 8.925879505394255456545714994997, 9.581011746947229235551958963245, 11.13170479928931682978453851607, 12.28502823723943026565793382675

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