Properties

Label 2-147-1.1-c5-0-11
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.44·2-s − 9·3-s − 2.33·4-s + 36·5-s − 49.0·6-s − 187.·8-s + 81·9-s + 196.·10-s + 184.·11-s + 21.0·12-s + 147.·13-s − 324·15-s − 943.·16-s + 1.96e3·17-s + 441.·18-s + 1.89e3·19-s − 84.1·20-s + 1.00e3·22-s + 136.·23-s + 1.68e3·24-s − 1.82e3·25-s + 805.·26-s − 729·27-s − 1.25e3·29-s − 1.76e3·30-s + 8.96e3·31-s + 844.·32-s + ⋯
L(s)  = 1  + 0.962·2-s − 0.577·3-s − 0.0730·4-s + 0.643·5-s − 0.555·6-s − 1.03·8-s + 0.333·9-s + 0.620·10-s + 0.459·11-s + 0.0421·12-s + 0.242·13-s − 0.371·15-s − 0.921·16-s + 1.65·17-s + 0.320·18-s + 1.20·19-s − 0.0470·20-s + 0.442·22-s + 0.0539·23-s + 0.596·24-s − 0.585·25-s + 0.233·26-s − 0.192·27-s − 0.278·29-s − 0.357·30-s + 1.67·31-s + 0.145·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.698660200\)
\(L(\frac12)\) \(\approx\) \(2.698660200\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
7 \( 1 \)
good2 \( 1 - 5.44T + 32T^{2} \)
5 \( 1 - 36T + 3.12e3T^{2} \)
11 \( 1 - 184.T + 1.61e5T^{2} \)
13 \( 1 - 147.T + 3.71e5T^{2} \)
17 \( 1 - 1.96e3T + 1.41e6T^{2} \)
19 \( 1 - 1.89e3T + 2.47e6T^{2} \)
23 \( 1 - 136.T + 6.43e6T^{2} \)
29 \( 1 + 1.25e3T + 2.05e7T^{2} \)
31 \( 1 - 8.96e3T + 2.86e7T^{2} \)
37 \( 1 - 1.28e4T + 6.93e7T^{2} \)
41 \( 1 - 8.97e3T + 1.15e8T^{2} \)
43 \( 1 - 1.35e4T + 1.47e8T^{2} \)
47 \( 1 + 2.00e4T + 2.29e8T^{2} \)
53 \( 1 - 9.33e3T + 4.18e8T^{2} \)
59 \( 1 - 8.86e3T + 7.14e8T^{2} \)
61 \( 1 + 4.11e4T + 8.44e8T^{2} \)
67 \( 1 + 5.53e4T + 1.35e9T^{2} \)
71 \( 1 + 6.38e4T + 1.80e9T^{2} \)
73 \( 1 - 4.12e4T + 2.07e9T^{2} \)
79 \( 1 - 1.69e4T + 3.07e9T^{2} \)
83 \( 1 + 1.01e5T + 3.93e9T^{2} \)
89 \( 1 - 8.71e4T + 5.58e9T^{2} \)
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

−12.19289985383651485720853556257, −11.54671236992937161041205407819, −10.04104152857849535960409107043, −9.323694854004559267706260818217, −7.73888855441177545298502215249, −6.17308660688735500903120277548, −5.58783053482362562003933596407, −4.39315914171940491356431612961, −3.06159944040201632589000446740, −1.04175397356406014468170844194, 1.04175397356406014468170844194, 3.06159944040201632589000446740, 4.39315914171940491356431612961, 5.58783053482362562003933596407, 6.17308660688735500903120277548, 7.73888855441177545298502215249, 9.323694854004559267706260818217, 10.04104152857849535960409107043, 11.54671236992937161041205407819, 12.19289985383651485720853556257

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