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Properties

Label 2-147-7.4-c3-0-16
Degree $2$
Conductor $147$
Sign $-0.386 + 0.922i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (1.5 − 2.59i)3-s + (3.5 − 6.06i)4-s + (−6 − 10.3i)5-s + 3·6-s + 15·8-s + (−4.5 − 7.79i)9-s + (6 − 10.3i)10-s + (−10 + 17.3i)11-s + (−10.5 − 18.1i)12-s − 84·13-s − 36·15-s + (−20.5 − 35.5i)16-s + (48 − 83.1i)17-s + (4.5 − 7.79i)18-s + (−6 − 10.3i)19-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (0.288 − 0.499i)3-s + (0.437 − 0.757i)4-s + (−0.536 − 0.929i)5-s + 0.204·6-s + 0.662·8-s + (−0.166 − 0.288i)9-s + (0.189 − 0.328i)10-s + (−0.274 + 0.474i)11-s + (−0.252 − 0.437i)12-s − 1.79·13-s − 0.619·15-s + (−0.320 − 0.554i)16-s + (0.684 − 1.18i)17-s + (0.0589 − 0.102i)18-s + (−0.0724 − 0.125i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.938378 - 1.41070i\)
\(L(\frac12)\) \(\approx\) \(0.938378 - 1.41070i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (6 + 10.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 84T + 2.19e3T^{2} \)
17 \( 1 + (-48 + 83.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6 + 10.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-88 - 152. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + (-132 + 228. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (129 + 223. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 156T + 7.95e4T^{2} \)
47 \( 1 + (-204 - 353. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-361 + 625. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (246 - 426. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-246 - 426. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (206 - 356. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 296T + 3.57e5T^{2} \)
73 \( 1 + (120 - 207. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 924T + 5.71e5T^{2} \)
89 \( 1 + (-372 - 644. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 168T + 9.12e5T^{2} \)
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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

−12.24102599810251475786184219010, −11.58208353125720225941805798782, −10.03627259599089634942180402394, −9.225042833989514938367429944162, −7.63775813588671663128222601371, −7.19304235928931771188543906366, −5.49843539741683864808213251907, −4.64626434302029699052319027147, −2.45786968484432981470674604666, −0.74290047392436893700699512415, 2.58161775498685181372486855816, 3.43461730909433756698421576553, 4.78428184717168352257230697998, 6.68549046020235602212343745084, 7.64317755299277191550512389863, 8.585223767735285287400664909796, 10.28203702064239206441556067341, 10.73664593773720509217393393141, 11.98256732539835819681854399140, 12.60565661471013685762165510206

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