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Properties

Label 2-147-7.4-c5-0-14
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  + (1 + 1.73i)2-s + (−4.5 + 7.79i)3-s + (14 − 24.2i)4-s + (5.5 + 9.52i)5-s − 18·6-s + 120·8-s + (−40.5 − 70.1i)9-s + (−11 + 19.0i)10-s + (−134.5 + 232. i)11-s + (126 + 218. i)12-s + 308·13-s − 99·15-s + (−328 − 568. i)16-s + (948 − 1.64e3i)17-s + (81 − 140. i)18-s + (−82 − 142. i)19-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (−0.288 + 0.499i)3-s + (0.437 − 0.757i)4-s + (0.0983 + 0.170i)5-s − 0.204·6-s + 0.662·8-s + (−0.166 − 0.288i)9-s + (−0.0347 + 0.0602i)10-s + (−0.335 + 0.580i)11-s + (0.252 + 0.437i)12-s + 0.505·13-s − 0.113·15-s + (−0.320 − 0.554i)16-s + (0.795 − 1.37i)17-s + (0.0589 − 0.102i)18-s + (−0.0521 − 0.0902i)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

\(L(3)\) \(\approx\) \(2.306668408\)
\(L(\frac12)\) \(\approx\) \(2.306668408\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-5.5 - 9.52i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (134.5 - 232. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 308T + 3.71e5T^{2} \)
17 \( 1 + (-948 + 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (82 + 142. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.63e3 - 2.82e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.41e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.42e3 + 2.46e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.66e3 - 9.81e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.68e4T + 1.15e8T^{2} \)
43 \( 1 + 7.89e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.05e4 - 1.82e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.48e4 + 2.57e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (4.08e3 - 7.06e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.58e3 - 1.31e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.60e4 + 2.77e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.82e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.74e4 + 3.01e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (6.76e3 + 1.17e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.81e4T + 3.93e9T^{2} \)
89 \( 1 + (5.74e4 + 9.95e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.54e5T + 8.58e9T^{2} \)
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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

−11.93783720507475785183272687242, −11.08676033047376916669334858246, −10.12645451035751111441606900188, −9.384454840357412981206886109539, −7.70703392405075898283136787696, −6.62934993544056546663085868975, −5.53297537157451948833787164565, −4.59292779060214025447775974507, −2.76703319405590425898550805371, −0.966686720871476818426598796120, 1.15097784376412864869139889781, 2.66517949136404440465002444918, 3.97719688013276235213310062876, 5.61827144270348898062444257039, 6.76070588265984703598428970669, 7.925991324269361223631647725320, 8.733143885899831737987035633082, 10.52835100839850734586373957683, 11.12891755145463211100796989564, 12.36184671546072524428835251581

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