Properties

Label 2-22848-1.1-c1-0-48
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 $1$

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 − 6·19-s − 21-s − 2·23-s − 4·25-s + 27-s − 6·29-s + 4·31-s − 3·33-s + 35-s + 11·37-s + 3·39-s + 12·41-s − 3·43-s − 45-s + 12·47-s + 49-s + 51-s − 5·53-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 − 1.37·19-s − 0.218·21-s − 0.417·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.522·33-s + 0.169·35-s + 1.80·37-s + 0.480·39-s + 1.87·41-s − 0.457·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s − 0.686·53-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: \(1\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−15.66087887309730, −15.38094884825137, −14.65593336121239, −14.21920817750323, −13.45341041862640, −13.12643436229254, −12.63963007916353, −12.07562253502179, −11.20923571576731, −10.91104579815153, −10.28149227049009, −9.579202823721551, −9.191636471914927, −8.337783373195654, −8.018090837407073, −7.532645902520769, −6.771436724937658, −5.967677953790337, −5.712867373120653, −4.531103966769734, −4.106263330003355, −3.523195836459961, −2.619116118583943, −2.185200029369879, −1.035229977024495, 0, 1.035229977024495, 2.185200029369879, 2.619116118583943, 3.523195836459961, 4.106263330003355, 4.531103966769734, 5.712867373120653, 5.967677953790337, 6.771436724937658, 7.532645902520769, 8.018090837407073, 8.337783373195654, 9.191636471914927, 9.579202823721551, 10.28149227049009, 10.91104579815153, 11.20923571576731, 12.07562253502179, 12.63963007916353, 13.12643436229254, 13.45341041862640, 14.21920817750323, 14.65593336121239, 15.38094884825137, 15.66087887309730

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