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Properties

Label 2-147-147.125-c1-0-5
Degree $2$
Conductor $147$
Sign $-0.406 - 0.913i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.880 + 1.82i)2-s + (0.750 + 1.56i)3-s + (−1.32 + 1.65i)4-s + (0.138 − 0.607i)5-s + (−2.19 + 2.74i)6-s + (−0.341 − 2.62i)7-s + (−0.237 − 0.0542i)8-s + (−1.87 + 2.34i)9-s + (1.23 − 0.281i)10-s + (−1.04 − 2.18i)11-s + (−3.57 − 0.820i)12-s + (−1.05 − 2.19i)13-s + (4.49 − 2.93i)14-s + (1.05 − 0.239i)15-s + (0.833 + 3.65i)16-s + (−2.77 − 3.47i)17-s + ⋯
L(s)  = 1  + (0.622 + 1.29i)2-s + (0.433 + 0.901i)3-s + (−0.660 + 0.828i)4-s + (0.0620 − 0.271i)5-s + (−0.895 + 1.12i)6-s + (−0.129 − 0.991i)7-s + (−0.0840 − 0.0191i)8-s + (−0.624 + 0.780i)9-s + (0.390 − 0.0890i)10-s + (−0.316 − 0.657i)11-s + (−1.03 − 0.236i)12-s + (−0.293 − 0.608i)13-s + (1.20 − 0.784i)14-s + (0.271 − 0.0618i)15-s + (0.208 + 0.912i)16-s + (−0.672 − 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.406 - 0.913i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888981 + 1.36813i\)
\(L(\frac12)\) \(\approx\) \(0.888981 + 1.36813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.750 - 1.56i)T \)
7 \( 1 + (0.341 + 2.62i)T \)
good2 \( 1 + (-0.880 - 1.82i)T + (-1.24 + 1.56i)T^{2} \)
5 \( 1 + (-0.138 + 0.607i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (1.04 + 2.18i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (1.05 + 2.19i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.77 + 3.47i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 1.89iT - 19T^{2} \)
23 \( 1 + (-2.16 - 1.72i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (4.37 - 3.48i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 - 1.50iT - 31T^{2} \)
37 \( 1 + (-6.15 - 7.72i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-1.46 + 6.40i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.34 + 10.2i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.26 - 2.53i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-6.80 - 5.42i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (3.31 + 14.5i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-0.689 + 0.550i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 5.03T + 67T^{2} \)
71 \( 1 + (-1.13 - 0.908i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (2.68 - 5.56i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (1.58 + 0.765i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-10.4 - 5.04i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 - 19.2iT - 97T^{2} \)
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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

−13.65563850243004238302610280157, −13.01511399102841267357379681684, −11.13898651295718322535410142569, −10.29719896388571958367389574457, −9.009613986199691370092518893145, −7.951324897655344130724367292505, −6.99148571813648649811233272503, −5.50688918474096225435372046916, −4.67770988433412401597391601551, −3.39630488888149246273474128842, 2.00701154220164908234991168942, 2.85158882010186307987503266274, 4.49765205163995628625680509875, 6.12453813479696201361935876146, 7.38690808594093391994602635004, 8.780738552711795280920426064947, 9.799057261820345284759277987479, 11.15309732853253740890923456268, 11.85915585066022563514039832885, 12.91653482010063811343202833163

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