Properties

Label 2-168-1.1-c3-0-5
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 + 20.3·5-s + 7·7-s + 9·9-s − 30.9·11-s + 50.6·13-s + 60.9·15-s − 102.·17-s − 61.2·19-s + 21·21-s + 148.·23-s + 287.·25-s + 27·27-s + 159.·29-s − 121.·31-s − 92.7·33-s + 142.·35-s − 357.·37-s + 151.·39-s + 466.·41-s − 185.·43-s + 182.·45-s − 131.·47-s + 49·49-s − 308.·51-s + 200.·53-s − 627.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.81·5-s + 0.377·7-s + 0.333·9-s − 0.847·11-s + 1.07·13-s + 1.04·15-s − 1.46·17-s − 0.739·19-s + 0.218·21-s + 1.34·23-s + 2.29·25-s + 0.192·27-s + 1.01·29-s − 0.702·31-s − 0.489·33-s + 0.686·35-s − 1.59·37-s + 0.623·39-s + 1.77·41-s − 0.658·43-s + 0.605·45-s − 0.407·47-s + 0.142·49-s − 0.846·51-s + 0.518·53-s − 1.53·55-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

\(L(2)\) \(\approx\) \(2.754563640\)
\(L(\frac12)\) \(\approx\) \(2.754563640\)
\(L(\frac{5}{2})\) 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 \( 1 - 20.3T + 125T^{2} \)
11 \( 1 + 30.9T + 1.33e3T^{2} \)
13 \( 1 - 50.6T + 2.19e3T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 + 61.2T + 6.85e3T^{2} \)
23 \( 1 - 148.T + 1.21e4T^{2} \)
29 \( 1 - 159.T + 2.43e4T^{2} \)
31 \( 1 + 121.T + 2.97e4T^{2} \)
37 \( 1 + 357.T + 5.06e4T^{2} \)
41 \( 1 - 466.T + 6.89e4T^{2} \)
43 \( 1 + 185.T + 7.95e4T^{2} \)
47 \( 1 + 131.T + 1.03e5T^{2} \)
53 \( 1 - 200.T + 1.48e5T^{2} \)
59 \( 1 + 591.T + 2.05e5T^{2} \)
61 \( 1 - 70.5T + 2.26e5T^{2} \)
67 \( 1 + 643.T + 3.00e5T^{2} \)
71 \( 1 + 522.T + 3.57e5T^{2} \)
73 \( 1 + 576.T + 3.89e5T^{2} \)
79 \( 1 - 280.T + 4.93e5T^{2} \)
83 \( 1 + 557.T + 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 65.0T + 9.12e5T^{2} \)
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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

−12.82968373319003569046052129900, −10.94651193254892048556377888356, −10.39378168409657923298955351062, −9.104576159802684794036175587543, −8.588666547680421905458836578486, −6.93989463786453034573750374663, −5.90653017967864764418028612321, −4.71298521037037433772872947045, −2.76376987435710596838630071607, −1.64662031906487988199545594164, 1.64662031906487988199545594164, 2.76376987435710596838630071607, 4.71298521037037433772872947045, 5.90653017967864764418028612321, 6.93989463786453034573750374663, 8.588666547680421905458836578486, 9.104576159802684794036175587543, 10.39378168409657923298955351062, 10.94651193254892048556377888356, 12.82968373319003569046052129900

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