Properties

Label 2-22848-1.1-c1-0-44
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 + 3·5-s + 7-s + 9-s + 3·11-s − 13-s − 3·15-s − 17-s + 7·19-s − 21-s + 23-s + 4·25-s − 27-s + 10·29-s + 4·31-s − 3·33-s + 3·35-s + 10·37-s + 39-s + 3·41-s + 11·43-s + 3·45-s − 8·47-s + 49-s + 51-s + 4·53-s + 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.774·15-s − 0.242·17-s + 1.60·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.522·33-s + 0.507·35-s + 1.64·37-s + 0.160·39-s + 0.468·41-s + 1.67·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.549·53-s + 1.21·55-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

\(L(1)\) \(\approx\) \(3.572216455\)
\(L(\frac12)\) \(\approx\) \(3.572216455\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.57829586672837, −14.84194509756459, −14.26870839949289, −13.78014882327901, −13.60910747339306, −12.59714926891574, −12.34327462109611, −11.57568395180430, −11.21680985736470, −10.52031424513650, −9.839309837622952, −9.573022714102662, −9.049978328185992, −8.223138620075912, −7.564779885681911, −6.875618699798949, −6.202691253748164, −5.978224328854640, −5.086338104985545, −4.734468432131598, −3.926976304599646, −2.880447738003235, −2.340902489858167, −1.274611651805787, −0.9406753868897240, 0.9406753868897240, 1.274611651805787, 2.340902489858167, 2.880447738003235, 3.926976304599646, 4.734468432131598, 5.086338104985545, 5.978224328854640, 6.202691253748164, 6.875618699798949, 7.564779885681911, 8.223138620075912, 9.049978328185992, 9.573022714102662, 9.839309837622952, 10.52031424513650, 11.21680985736470, 11.57568395180430, 12.34327462109611, 12.59714926891574, 13.60910747339306, 13.78014882327901, 14.26870839949289, 14.84194509756459, 15.57829586672837

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