Properties

Label 2-22848-1.1-c1-0-12
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 + 11-s − 3·13-s − 3·15-s + 17-s + 6·19-s + 21-s − 2·23-s + 4·25-s + 27-s − 6·29-s + 33-s − 3·35-s − 3·37-s − 3·39-s − 11·43-s − 3·45-s + 49-s + 51-s + 9·53-s − 3·55-s + 6·57-s + 4·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.774·15-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 + 0.174·33-s − 0.507·35-s − 0.493·37-s − 0.480·39-s − 1.67·43-s − 0.447·45-s + 1/7·49-s + 0.140·51-s + 1.23·53-s − 0.404·55-s + 0.794·57-s + 0.520·59-s + 1.79·61-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: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.789776809\)
\(L(\frac12)\) \(\approx\) \(1.789776809\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
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 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−15.37500485692013, −14.94919633959364, −14.53890484101258, −13.96505331606904, −13.39566234553318, −12.70919134535956, −12.10833364643383, −11.67854337798926, −11.36757037626729, −10.55848754685265, −9.806084199324646, −9.510874338537468, −8.652460601468433, −8.115946597034118, −7.791715688834880, −7.015651735053542, −6.868593547856916, −5.464893348288985, −5.220635855620681, −4.247674769285046, −3.799281746726978, −3.242469894758888, −2.408687915674638, −1.535057818319933, −0.5304375824482973, 0.5304375824482973, 1.535057818319933, 2.408687915674638, 3.242469894758888, 3.799281746726978, 4.247674769285046, 5.220635855620681, 5.464893348288985, 6.868593547856916, 7.015651735053542, 7.791715688834880, 8.115946597034118, 8.652460601468433, 9.510874338537468, 9.806084199324646, 10.55848754685265, 11.36757037626729, 11.67854337798926, 12.10833364643383, 12.70919134535956, 13.39566234553318, 13.96505331606904, 14.53890484101258, 14.94919633959364, 15.37500485692013

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