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Properties

Label 2-147-147.104-c1-0-11
Degree $2$
Conductor $147$
Sign $-0.0106 + 0.999i$
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  + (0.316 + 0.252i)2-s + (−1.66 + 0.476i)3-s + (−0.408 − 1.78i)4-s + (−1.71 − 0.827i)5-s + (−0.647 − 0.269i)6-s + (1.33 − 2.28i)7-s + (0.674 − 1.40i)8-s + (2.54 − 1.58i)9-s + (−0.335 − 0.695i)10-s + (−3.02 − 2.41i)11-s + (1.53 + 2.78i)12-s + (0.165 + 0.131i)13-s + (1.00 − 0.386i)14-s + (3.25 + 0.559i)15-s + (−2.74 + 1.31i)16-s + (−0.675 + 2.96i)17-s + ⋯
L(s)  = 1  + (0.223 + 0.178i)2-s + (−0.961 + 0.274i)3-s + (−0.204 − 0.894i)4-s + (−0.768 − 0.369i)5-s + (−0.264 − 0.110i)6-s + (0.504 − 0.863i)7-s + (0.238 − 0.495i)8-s + (0.848 − 0.528i)9-s + (−0.105 − 0.220i)10-s + (−0.912 − 0.727i)11-s + (0.442 + 0.804i)12-s + (0.0457 + 0.0365i)13-s + (0.267 − 0.103i)14-s + (0.840 + 0.144i)15-s + (−0.685 + 0.329i)16-s + (−0.163 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.0106 + 0.999i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.0106 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500185 - 0.505521i\)
\(L(\frac12)\) \(\approx\) \(0.500185 - 0.505521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 - 0.476i)T \)
7 \( 1 + (-1.33 + 2.28i)T \)
good2 \( 1 + (-0.316 - 0.252i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.71 + 0.827i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (3.02 + 2.41i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.165 - 0.131i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.675 - 2.96i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 0.353iT - 19T^{2} \)
23 \( 1 + (-5.70 + 1.30i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-4.53 - 1.03i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.53iT - 31T^{2} \)
37 \( 1 + (1.82 - 7.97i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-6.08 - 2.92i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.39 + 2.59i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-6.89 + 8.64i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-7.60 + 1.73i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (7.53 - 3.63i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.504 - 0.115i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 + (12.0 - 2.74i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.11 + 2.48i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 3.31T + 79T^{2} \)
83 \( 1 + (9.52 + 11.9i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.22 - 1.53i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−12.87947921460535745308968403816, −11.58669725454969427382045262287, −10.75651933207651257424222607047, −10.16197544028052720213920242558, −8.601666523387302813962746059802, −7.30955224656196422825624856862, −6.05523453089131507767216008856, −4.94832672778531998354674423466, −4.09958044909231118907071994990, −0.77397696886767567295232011039, 2.64963554106249568055633644327, 4.43788721448293254564867732582, 5.39577201586744268759473108009, 7.15964092744073831288287639348, 7.76086971498589664224127528704, 9.094075146088328897329528119344, 10.77152172877333819213539351787, 11.44705896021230059409809213437, 12.32775495768211235578228242332, 12.83610605095033525582595183044

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