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Properties

Label 2-147-21.20-c5-0-59
Degree $2$
Conductor $147$
Sign $-0.268 - 0.963i$
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  − 6.54i·2-s + (2.32 − 15.4i)3-s − 10.7·4-s + 13.6·5-s + (−100. − 15.2i)6-s − 138. i·8-s + (−232. − 71.7i)9-s − 89.4i·10-s + 214. i·11-s + (−25.0 + 166. i)12-s − 219. i·13-s + (31.8 − 210. i)15-s − 1.25e3·16-s − 2.12e3·17-s + (−469. + 1.51e3i)18-s + 1.22e3i·19-s + ⋯
L(s)  = 1  − 1.15i·2-s + (0.149 − 0.988i)3-s − 0.336·4-s + 0.244·5-s + (−1.14 − 0.172i)6-s − 0.766i·8-s + (−0.955 − 0.295i)9-s − 0.283i·10-s + 0.535i·11-s + (−0.0502 + 0.332i)12-s − 0.360i·13-s + (0.0365 − 0.242i)15-s − 1.22·16-s − 1.78·17-s + (−0.341 + 1.10i)18-s + 0.775i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.268 - 0.963i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.153144225\)
\(L(\frac12)\) \(\approx\) \(1.153144225\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.32 + 15.4i)T \)
7 \( 1 \)
good2 \( 1 + 6.54iT - 32T^{2} \)
5 \( 1 - 13.6T + 3.12e3T^{2} \)
11 \( 1 - 214. iT - 1.61e5T^{2} \)
13 \( 1 + 219. iT - 3.71e5T^{2} \)
17 \( 1 + 2.12e3T + 1.41e6T^{2} \)
19 \( 1 - 1.22e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.30e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.63e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.58e3iT - 2.86e7T^{2} \)
37 \( 1 + 373.T + 6.93e7T^{2} \)
41 \( 1 - 1.09e4T + 1.15e8T^{2} \)
43 \( 1 - 2.07e4T + 1.47e8T^{2} \)
47 \( 1 + 1.51e4T + 2.29e8T^{2} \)
53 \( 1 + 2.43e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.90e4T + 7.14e8T^{2} \)
61 \( 1 - 1.40e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.45e4T + 1.35e9T^{2} \)
71 \( 1 + 4.15e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.86e3iT - 2.07e9T^{2} \)
79 \( 1 - 3.39e4T + 3.07e9T^{2} \)
83 \( 1 - 9.96e3T + 3.93e9T^{2} \)
89 \( 1 + 1.15e5T + 5.58e9T^{2} \)
97 \( 1 + 9.88e4iT - 8.58e9T^{2} \)
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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

−11.53957158748388923150813114016, −10.63866717393373598294874651481, −9.541204180825326820108905758550, −8.374782232640140520863777847738, −7.06875151531422170671667058372, −6.10857570266145925317946725170, −4.21714679461866388321060972508, −2.62876482321242168049844050555, −1.82594267930100472635486296932, −0.34824832863185983423521746707, 2.44573729695985547158941251516, 4.17264830270449494109503085722, 5.35533537280678791220246398599, 6.30756017308596023347277218007, 7.54337417406247973274443052972, 8.771399931418310808326760439337, 9.392736361360494652826155949375, 10.94844241116708777733773592910, 11.44174575316294005126156863451, 13.34710178217964724282317052228

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