Properties

Label 2-22848-1.1-c1-0-49
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 − 2·5-s − 7-s + 9-s + 2·13-s − 2·15-s − 17-s + 6·19-s − 21-s − 8·23-s − 25-s + 27-s + 4·29-s + 2·35-s − 10·37-s + 2·39-s + 10·41-s − 8·43-s − 2·45-s + 8·47-s + 49-s − 51-s − 6·53-s + 6·57-s − 4·59-s + 8·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.242·17-s + 1.37·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.338·35-s − 1.64·37-s + 0.320·39-s + 1.56·41-s − 1.21·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 0.794·57-s − 0.520·59-s + 1.02·61-s − 0.125·63-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.75023863758487, −15.52574502551944, −14.58807776264534, −14.06613321633671, −13.73779171205936, −13.15851830784260, −12.25760127624452, −12.15067816279963, −11.46596977942206, −10.86404828420864, −10.14753841786253, −9.737874217874475, −9.021376332527187, −8.515710477018588, −7.833985749308928, −7.578808392157771, −6.783127811277679, −6.190609797297254, −5.449137902263404, −4.694610628651968, −3.831109865674417, −3.648393866565273, −2.828459088903623, −2.003829126729644, −1.051185241688346, 0, 1.051185241688346, 2.003829126729644, 2.828459088903623, 3.648393866565273, 3.831109865674417, 4.694610628651968, 5.449137902263404, 6.190609797297254, 6.783127811277679, 7.578808392157771, 7.833985749308928, 8.515710477018588, 9.021376332527187, 9.737874217874475, 10.14753841786253, 10.86404828420864, 11.46596977942206, 12.15067816279963, 12.25760127624452, 13.15851830784260, 13.73779171205936, 14.06613321633671, 14.58807776264534, 15.52574502551944, 15.75023863758487

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