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Properties

Label 2-147-7.2-c5-0-11
Degree $2$
Conductor $147$
Sign $-0.386 - 0.922i$
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  + (−2.72 + 4.71i)2-s + (−4.5 − 7.79i)3-s + (1.16 + 2.02i)4-s + (18 − 31.1i)5-s + 49.0·6-s − 187.·8-s + (−40.5 + 70.1i)9-s + (98.0 + 169. i)10-s + (−92.2 − 159. i)11-s + (10.5 − 18.2i)12-s − 147.·13-s − 324·15-s + (471. − 817. i)16-s + (984. + 1.70e3i)17-s + (−220. − 382. i)18-s + (946. − 1.63e3i)19-s + ⋯
L(s)  = 1  + (−0.481 + 0.833i)2-s + (−0.288 − 0.499i)3-s + (0.0365 + 0.0632i)4-s + (0.321 − 0.557i)5-s + 0.555·6-s − 1.03·8-s + (−0.166 + 0.288i)9-s + (0.310 + 0.536i)10-s + (−0.229 − 0.397i)11-s + (0.0210 − 0.0365i)12-s − 0.242·13-s − 0.371·15-s + (0.460 − 0.798i)16-s + (0.825 + 1.43i)17-s + (−0.160 − 0.277i)18-s + (0.601 − 1.04i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
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: \((2,\ 147,\ (\ :5/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.074914322\)
\(L(\frac12)\) \(\approx\) \(1.074914322\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.72 - 4.71i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-18 + 31.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (92.2 + 159. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 147.T + 3.71e5T^{2} \)
17 \( 1 + (-984. - 1.70e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-946. + 1.63e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (68.4 - 118. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 1.25e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.48e3 - 7.76e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (6.44e3 - 1.11e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 8.97e3T + 1.15e8T^{2} \)
43 \( 1 - 1.35e4T + 1.47e8T^{2} \)
47 \( 1 + (1.00e4 - 1.73e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (4.66e3 + 8.08e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-4.43e3 - 7.67e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.05e4 - 3.56e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.76e4 - 4.79e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.38e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.06e4 - 3.57e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (8.48e3 - 1.46e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.01e5T + 3.93e9T^{2} \)
89 \( 1 + (-4.35e4 + 7.54e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.18e5T + 8.58e9T^{2} \)
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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

−12.50269142519634616817399501587, −11.62303347943035976234938067510, −10.28318524510038007904870296454, −8.979475802117528928242096572000, −8.206762316558817103292743599757, −7.18036492138922651352710120275, −6.15056555912752834632810224665, −5.13393218386062056853210059318, −3.08689152358515393756691772641, −1.19366849530245403822619531518, 0.49244682847355574768995362604, 2.18324051894513451154729981862, 3.38238524204712429410322952491, 5.16396460015153211399873043696, 6.26971477684641125710902287419, 7.65275987164542417806405588247, 9.284705836750548717170758606941, 9.944306621427415484763492560506, 10.62855220372190781064662122991, 11.68372654581346772852129316448

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