Properties

Label 2-22848-1.1-c1-0-51
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 − 6·11-s + 4·13-s − 2·15-s + 17-s − 6·19-s − 21-s + 4·23-s − 25-s − 27-s + 8·29-s + 6·33-s + 2·35-s − 4·39-s − 2·41-s − 4·43-s + 2·45-s + 49-s − 51-s − 2·53-s − 12·55-s + 6·57-s − 6·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.516·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.04·33-s + 0.338·35-s − 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.140·51-s − 0.274·53-s − 1.61·55-s + 0.794·57-s − 0.781·59-s − 0.256·61-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−15.61977282147934, −15.44048265485278, −14.72758941395874, −13.95670406451197, −13.51250754030126, −13.12020072942732, −12.60026965201582, −12.02975784675591, −11.11370500440661, −10.87992828124500, −10.29251909057437, −9.979484719254681, −9.086461374646846, −8.436865297109473, −8.068820512381798, −7.295890399183400, −6.501149724840218, −6.093826302198448, −5.478402812961354, −4.921840819014411, −4.393901601860455, −3.353900529932238, −2.612759483417038, −1.914674082892808, −1.102062505337669, 0, 1.102062505337669, 1.914674082892808, 2.612759483417038, 3.353900529932238, 4.393901601860455, 4.921840819014411, 5.478402812961354, 6.093826302198448, 6.501149724840218, 7.295890399183400, 8.068820512381798, 8.436865297109473, 9.086461374646846, 9.979484719254681, 10.29251909057437, 10.87992828124500, 11.11370500440661, 12.02975784675591, 12.60026965201582, 13.12020072942732, 13.51250754030126, 13.95670406451197, 14.72758941395874, 15.44048265485278, 15.61977282147934

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