Properties

Label 2-168-1.1-c3-0-3
Degree $2$
Conductor $168$
Sign $1$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 6.30·5-s + 7·7-s + 9·9-s + 48.9·11-s − 2.60·13-s − 18.9·15-s + 136.·17-s + 45.2·19-s + 21·21-s − 38.1·23-s − 85.2·25-s + 27·27-s + 52.7·29-s − 14.7·31-s + 146.·33-s − 44.1·35-s + 333.·37-s − 7.82·39-s + 227.·41-s − 398.·43-s − 56.7·45-s − 184.·47-s + 49·49-s + 410.·51-s + 359.·53-s − 308.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.563·5-s + 0.377·7-s + 0.333·9-s + 1.34·11-s − 0.0556·13-s − 0.325·15-s + 1.95·17-s + 0.545·19-s + 0.218·21-s − 0.345·23-s − 0.682·25-s + 0.192·27-s + 0.337·29-s − 0.0856·31-s + 0.774·33-s − 0.213·35-s + 1.48·37-s − 0.0321·39-s + 0.865·41-s − 1.41·43-s − 0.187·45-s − 0.572·47-s + 0.142·49-s + 1.12·51-s + 0.932·53-s − 0.755·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.141246333\)
\(L(\frac12)\) \(\approx\) \(2.141246333\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 - 7T \)
good5 \( 1 + 6.30T + 125T^{2} \)
11 \( 1 - 48.9T + 1.33e3T^{2} \)
13 \( 1 + 2.60T + 2.19e3T^{2} \)
17 \( 1 - 136.T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 + 38.1T + 1.21e4T^{2} \)
29 \( 1 - 52.7T + 2.43e4T^{2} \)
31 \( 1 + 14.7T + 2.97e4T^{2} \)
37 \( 1 - 333.T + 5.06e4T^{2} \)
41 \( 1 - 227.T + 6.89e4T^{2} \)
43 \( 1 + 398.T + 7.95e4T^{2} \)
47 \( 1 + 184.T + 1.03e5T^{2} \)
53 \( 1 - 359.T + 1.48e5T^{2} \)
59 \( 1 - 99.9T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 + 376.T + 3.00e5T^{2} \)
71 \( 1 + 1.18e3T + 3.57e5T^{2} \)
73 \( 1 + 735.T + 3.89e5T^{2} \)
79 \( 1 + 836.T + 4.93e5T^{2} \)
83 \( 1 - 293.T + 5.71e5T^{2} \)
89 \( 1 - 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 201.T + 9.12e5T^{2} \)
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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

−12.06888665870260186297538466713, −11.65708535942331082089134661137, −10.16300705707623329432265438692, −9.275007092314261546216238069534, −8.109065288083680973331230213696, −7.36519219012300296816301509798, −5.90615384353472700633124666286, −4.32798627258125771807911710005, −3.25237628222443020654370158887, −1.31445216384837415690922545194, 1.31445216384837415690922545194, 3.25237628222443020654370158887, 4.32798627258125771807911710005, 5.90615384353472700633124666286, 7.36519219012300296816301509798, 8.109065288083680973331230213696, 9.275007092314261546216238069534, 10.16300705707623329432265438692, 11.65708535942331082089134661137, 12.06888665870260186297538466713

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