Properties

Label 2-22848-1.1-c1-0-42
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 + 8·19-s + 21-s − 25-s − 27-s − 2·29-s + 2·35-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 2·45-s − 8·47-s + 49-s + 51-s + 6·53-s − 8·57-s − 4·59-s − 6·61-s − 63-s − 4·65-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 + 1.83·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.338·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s − 1.05·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.496·65-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 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.83772010194157, −15.36261747393649, −14.84152749951876, −14.04018236781743, −13.54796432841445, −13.07061193405109, −12.30599009186710, −11.95236196894772, −11.39188066033593, −11.04582925996724, −10.29869435504879, −9.725150178370204, −9.179221944530906, −8.507188063651212, −7.736519048628292, −7.429292355547449, −6.742968342238531, −6.060547785215804, −5.506082687282539, −4.815298649854199, −4.128023931299655, −3.471850991606325, −2.960522368968260, −1.765992212180782, −0.9105024180812468, 0, 0.9105024180812468, 1.765992212180782, 2.960522368968260, 3.471850991606325, 4.128023931299655, 4.815298649854199, 5.506082687282539, 6.060547785215804, 6.742968342238531, 7.429292355547449, 7.736519048628292, 8.507188063651212, 9.179221944530906, 9.725150178370204, 10.29869435504879, 11.04582925996724, 11.39188066033593, 11.95236196894772, 12.30599009186710, 13.07061193405109, 13.54796432841445, 14.04018236781743, 14.84152749951876, 15.36261747393649, 15.83772010194157

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