Properties

Label 2-22848-1.1-c1-0-5
Degree $2$
Conductor $22848$
Sign $1$
Analytic cond. $182.442$
Root an. cond. $13.5071$
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 − 5-s + 7-s + 9-s + 3·11-s − 3·13-s + 15-s + 17-s + 3·19-s − 21-s − 7·23-s − 4·25-s − 27-s + 6·29-s − 10·31-s − 3·33-s − 35-s − 4·37-s + 3·39-s − 9·41-s + 9·43-s − 45-s − 6·47-s + 49-s − 51-s + 10·53-s − 3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.832·13-s + 0.258·15-s + 0.242·17-s + 0.688·19-s − 0.218·21-s − 1.45·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.522·33-s − 0.169·35-s − 0.657·37-s + 0.480·39-s − 1.40·41-s + 1.37·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.140·51-s + 1.37·53-s − 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22848\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(182.442\)
Root analytic conductor: \(13.5071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171727308\)
\(L(\frac12)\) \(\approx\) \(1.171727308\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.58616156414899, −14.86424106294025, −14.50193120375127, −13.86451901312900, −13.46319169212251, −12.42822576844366, −12.12569311644467, −11.85177167916487, −11.20752506655161, −10.61832127973773, −9.939876350111053, −9.564236865059259, −8.817235756162750, −8.148847361652906, −7.563433601285031, −7.076028451320962, −6.438635577808265, −5.670362678001053, −5.244356017875472, −4.422653658628404, −3.918400181519161, −3.254815316308067, −2.144350105400330, −1.518408650232620, −0.4572627927578034, 0.4572627927578034, 1.518408650232620, 2.144350105400330, 3.254815316308067, 3.918400181519161, 4.422653658628404, 5.244356017875472, 5.670362678001053, 6.438635577808265, 7.076028451320962, 7.563433601285031, 8.148847361652906, 8.817235756162750, 9.564236865059259, 9.939876350111053, 10.61832127973773, 11.20752506655161, 11.85177167916487, 12.12569311644467, 12.42822576844366, 13.46319169212251, 13.86451901312900, 14.50193120375127, 14.86424106294025, 15.58616156414899

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