Properties

Label 2-147-1.1-c1-0-4
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s − 2·11-s − 2·12-s − 13-s − 2·15-s − 4·16-s + 2·18-s − 19-s + 4·20-s − 4·22-s − 25-s − 2·26-s − 27-s + 4·29-s − 4·30-s − 9·31-s − 8·32-s + 2·33-s + 2·36-s + 3·37-s − 2·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s − 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s − 0.229·19-s + 0.894·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.742·29-s − 0.730·30-s − 1.61·31-s − 1.41·32-s + 0.348·33-s + 1/3·36-s + 0.493·37-s − 0.324·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920053745\)
\(L(\frac12)\) \(\approx\) \(1.920053745\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.05041447442840207720378708897, −12.40866641446903147774840263019, −11.30162828605416851531895428330, −10.28136035144373451283122510308, −9.092563983560312555964020990816, −7.29485403540511127428179965078, −6.01621490934601452163404663718, −5.40855511243246313730427714163, −4.18739539622885934722496613081, −2.47062022887811000287520417928, 2.47062022887811000287520417928, 4.18739539622885934722496613081, 5.40855511243246313730427714163, 6.01621490934601452163404663718, 7.29485403540511127428179965078, 9.092563983560312555964020990816, 10.28136035144373451283122510308, 11.30162828605416851531895428330, 12.40866641446903147774840263019, 13.05041447442840207720378708897

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