Properties

Label 2-22848-1.1-c1-0-28
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 + 5-s − 7-s + 9-s + 11-s + 3·13-s + 15-s + 17-s − 5·19-s − 21-s + 7·23-s − 4·25-s + 27-s + 6·29-s + 4·31-s + 33-s − 35-s + 2·37-s + 3·39-s − 3·41-s + 7·43-s + 45-s − 4·47-s + 49-s + 51-s + 55-s − 5·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.258·15-s + 0.242·17-s − 1.14·19-s − 0.218·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.480·39-s − 0.468·41-s + 1.06·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.134·55-s − 0.662·57-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.450096824\)
\(L(\frac12)\) \(\approx\) \(3.450096824\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−15.60738179711590, −14.78885316992937, −14.51704477704421, −13.75918224789688, −13.36731573934866, −12.96337683596087, −12.32104577261678, −11.72696945960596, −10.97350004017310, −10.56213253342330, −9.850677415231462, −9.465131952107521, −8.679363400392526, −8.469460162572450, −7.709326015781265, −6.902025685257077, −6.467671654260723, −5.918349284386895, −5.122212304165880, −4.354909081633755, −3.786168100481584, −3.012559452657830, −2.431800163124788, −1.550333723950236, −0.7546684011825471, 0.7546684011825471, 1.550333723950236, 2.431800163124788, 3.012559452657830, 3.786168100481584, 4.354909081633755, 5.122212304165880, 5.918349284386895, 6.467671654260723, 6.902025685257077, 7.709326015781265, 8.469460162572450, 8.679363400392526, 9.465131952107521, 9.850677415231462, 10.56213253342330, 10.97350004017310, 11.72696945960596, 12.32104577261678, 12.96337683596087, 13.36731573934866, 13.75918224789688, 14.51704477704421, 14.78885316992937, 15.60738179711590

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