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Properties

Label 2-147-147.104-c1-0-12
Degree $2$
Conductor $147$
Sign $0.833 + 0.551i$
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  + (0.853 + 0.681i)2-s + (−0.522 − 1.65i)3-s + (−0.179 − 0.786i)4-s + (2.05 + 0.987i)5-s + (0.678 − 1.76i)6-s + (−1.78 − 1.95i)7-s + (1.33 − 2.76i)8-s + (−2.45 + 1.72i)9-s + (1.07 + 2.24i)10-s + (0.472 + 0.377i)11-s + (−1.20 + 0.707i)12-s + (4.60 + 3.67i)13-s + (−0.194 − 2.88i)14-s + (0.559 − 3.90i)15-s + (1.56 − 0.752i)16-s + (−1.04 + 4.57i)17-s + ⋯
L(s)  = 1  + (0.603 + 0.481i)2-s + (−0.301 − 0.953i)3-s + (−0.0897 − 0.393i)4-s + (0.917 + 0.441i)5-s + (0.277 − 0.720i)6-s + (−0.674 − 0.738i)7-s + (0.470 − 0.976i)8-s + (−0.818 + 0.575i)9-s + (0.341 + 0.708i)10-s + (0.142 + 0.113i)11-s + (−0.347 + 0.204i)12-s + (1.27 + 1.01i)13-s + (−0.0520 − 0.770i)14-s + (0.144 − 1.00i)15-s + (0.390 − 0.188i)16-s + (−0.253 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.833 + 0.551i$
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: \((2,\ 147,\ (\ :1/2),\ 0.833 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36277 - 0.410092i\)
\(L(\frac12)\) \(\approx\) \(1.36277 - 0.410092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.522 + 1.65i)T \)
7 \( 1 + (1.78 + 1.95i)T \)
good2 \( 1 + (-0.853 - 0.681i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.05 - 0.987i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-0.472 - 0.377i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.60 - 3.67i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.04 - 4.57i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 1.41iT - 19T^{2} \)
23 \( 1 + (5.00 - 1.14i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.134 - 0.0307i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 9.28iT - 31T^{2} \)
37 \( 1 + (-0.542 + 2.37i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (8.06 + 3.88i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.18 + 1.53i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.12 + 3.92i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-11.1 + 2.54i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.375 + 0.180i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.90 - 0.891i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (-1.75 + 0.401i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.31 + 4.24i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 8.03T + 79T^{2} \)
83 \( 1 + (-4.94 - 6.19i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (8.75 + 10.9i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 4.10iT - 97T^{2} \)
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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

−13.55353131302618070599710065565, −12.29514335249226634942457847230, −10.83645244681032199410076757926, −10.12547818856971332378564544610, −8.741057523940386148927271101664, −6.99922797784398789912189341294, −6.45804214877219285692363736629, −5.68770423555721068777701443127, −3.92258980409942910566362195652, −1.65741752992665387634843050274, 2.73166595703387638949202493383, 3.95568901560341340478774858891, 5.37177660046863120985277543308, 6.05105592519435484324530538663, 8.273129682449334211772280708019, 9.215613736502778361361454617711, 10.11794038535787419204719235431, 11.30948524272940170879706547770, 12.13129365257356627842569392650, 13.21346378931535852177172882570

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