Properties

Label 2-168-1.1-c1-0-1
Degree $2$
Conductor $168$
Sign $1$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
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  + 3-s + 2·5-s − 7-s + 9-s − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 21-s − 4·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s − 2·39-s − 10·41-s + 12·43-s + 2·45-s − 8·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s + 4·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451262370\)
\(L(\frac12)\) \(\approx\) \(1.451262370\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.82992732310357645044415807593, −12.06652674974971659339169474440, −10.42205409739275983523104171923, −9.848554569012739739807460123677, −8.820314983809012661425051705328, −7.65994621019403063529610936786, −6.42136658293515809263418102895, −5.25019213638990204722565722077, −3.56921408081238054859871707771, −2.06515692945354055956624655608, 2.06515692945354055956624655608, 3.56921408081238054859871707771, 5.25019213638990204722565722077, 6.42136658293515809263418102895, 7.65994621019403063529610936786, 8.820314983809012661425051705328, 9.848554569012739739807460123677, 10.42205409739275983523104171923, 12.06652674974971659339169474440, 12.82992732310357645044415807593

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