Properties

Label 2-22848-1.1-c1-0-2
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 − 2·5-s − 7-s + 9-s − 6·13-s − 2·15-s − 17-s + 2·19-s − 21-s − 25-s + 27-s − 8·29-s + 2·35-s − 2·37-s − 6·39-s + 2·41-s − 8·43-s − 2·45-s − 8·47-s + 49-s − 51-s + 2·53-s + 2·57-s − 12·59-s + 4·61-s − 63-s + 12·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.338·35-s − 0.328·37-s − 0.960·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.274·53-s + 0.264·57-s − 1.56·59-s + 0.512·61-s − 0.125·63-s + 1.48·65-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: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7877691491\)
\(L(\frac12)\) \(\approx\) \(0.7877691491\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.27656874475947, −15.07427557757760, −14.51447772039881, −13.93225899151469, −13.28679893178056, −12.76902617286011, −12.27120954626000, −11.65489174729174, −11.32690860510097, −10.39342879525033, −9.941117092709411, −9.375711926761354, −8.894785539602157, −8.114001857572779, −7.527314029012330, −7.331143091179387, −6.570869613537444, −5.766367010526887, −4.926237077770444, −4.516665327493326, −3.608521516159327, −3.246646512422803, −2.379553450754947, −1.682462395815057, −0.3292801843893025, 0.3292801843893025, 1.682462395815057, 2.379553450754947, 3.246646512422803, 3.608521516159327, 4.516665327493326, 4.926237077770444, 5.766367010526887, 6.570869613537444, 7.331143091179387, 7.527314029012330, 8.114001857572779, 8.894785539602157, 9.375711926761354, 9.941117092709411, 10.39342879525033, 11.32690860510097, 11.65489174729174, 12.27120954626000, 12.76902617286011, 13.28679893178056, 13.93225899151469, 14.51447772039881, 15.07427557757760, 15.27656874475947

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