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Properties

Label 2-147-7.2-c5-0-32
Degree $2$
Conductor $147$
Sign $0.0725 - 0.997i$
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  + (5.59 − 9.68i)2-s + (−4.5 − 7.79i)3-s + (−46.5 − 80.6i)4-s + (31.1 − 53.9i)5-s − 100.·6-s − 684.·8-s + (−40.5 + 70.1i)9-s + (−348. − 603. i)10-s + (86.7 + 150. i)11-s + (−419. + 726. i)12-s + 348.·13-s − 560.·15-s + (−2.33e3 + 4.04e3i)16-s + (−499. − 865. i)17-s + (453. + 784. i)18-s + (1.02e3 − 1.76e3i)19-s + ⋯
L(s)  = 1  + (0.988 − 1.71i)2-s + (−0.288 − 0.499i)3-s + (−1.45 − 2.52i)4-s + (0.557 − 0.965i)5-s − 1.14·6-s − 3.78·8-s + (−0.166 + 0.288i)9-s + (−1.10 − 1.90i)10-s + (0.216 + 0.374i)11-s + (−0.840 + 1.45i)12-s + 0.571·13-s − 0.643·15-s + (−2.28 + 3.95i)16-s + (−0.419 − 0.726i)17-s + (0.329 + 0.570i)18-s + (0.648 − 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0725 - 0.997i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.380831823\)
\(L(\frac12)\) \(\approx\) \(2.380831823\)
\(L(\frac{7}{2})\) 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 \( 1 + (-5.59 + 9.68i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-31.1 + 53.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-86.7 - 150. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 348.T + 3.71e5T^{2} \)
17 \( 1 + (499. + 865. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.02e3 + 1.76e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-974. + 1.68e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 738.T + 2.05e7T^{2} \)
31 \( 1 + (-1.22e3 - 2.13e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.39e3 - 7.60e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.76e4T + 1.15e8T^{2} \)
43 \( 1 + 1.03e4T + 1.47e8T^{2} \)
47 \( 1 + (-37.5 + 65.0i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.38e4 + 2.39e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.72e3 - 2.99e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (7.97e3 - 1.38e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.20e3 + 1.24e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.03e4T + 1.80e9T^{2} \)
73 \( 1 + (4.29e4 + 7.43e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.80e3 - 6.59e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 6.37e4T + 3.93e9T^{2} \)
89 \( 1 + (2.31e4 - 4.01e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 2.14e4T + 8.58e9T^{2} \)
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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

−11.62397669134791266925983864821, −10.72195695666922424731065482926, −9.530301495701188002664715167665, −8.822790848401448358503470844271, −6.51530589652247273709231658080, −5.25581294895005671777997891488, −4.57531692641465572326360621396, −2.89110194082055634373834967862, −1.58479163537371055861875116250, −0.63018685655651505440645101612, 3.17108339416318791120112167701, 4.17724153770335832495900387693, 5.67475185160422038083494641625, 6.17189938282955055411506220385, 7.25502039907712483677879455283, 8.428822182675078537997547439059, 9.574127595374352554810369883588, 11.00335621706847653615970873178, 12.24358797338193920223720307636, 13.36397142219203010686764925175

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