Properties

Label 2-22848-1.1-c1-0-32
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 + 5·11-s − 13-s + 3·15-s − 17-s − 3·19-s − 21-s − 5·23-s + 4·25-s + 27-s + 2·29-s + 5·33-s − 3·35-s − 2·37-s − 39-s + 7·41-s − 3·43-s + 3·45-s + 12·47-s + 49-s − 51-s − 8·53-s + 15·55-s − 3·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s − 0.688·19-s − 0.218·21-s − 1.04·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s + 0.870·33-s − 0.507·35-s − 0.328·37-s − 0.160·39-s + 1.09·41-s − 0.457·43-s + 0.447·45-s + 1.75·47-s + 1/7·49-s − 0.140·51-s − 1.09·53-s + 2.02·55-s − 0.397·57-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: $\chi_{22848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.054524015\)
\(L(\frac12)\) \(\approx\) \(4.054524015\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−15.49843059955769, −14.72012021690504, −14.28105283441123, −13.98661233722294, −13.47372971928918, −12.81363939328449, −12.37076210070121, −11.78749265611220, −11.02084479207643, −10.36762019643803, −9.836015744506314, −9.448222347339062, −8.915513795267085, −8.471653548441130, −7.583678680881843, −6.937104404765398, −6.224049534205340, −6.105581156641971, −5.183196509724414, −4.319109148517318, −3.858666966566378, −2.989134543959161, −2.171083853879936, −1.785714801761716, −0.7869294560334914, 0.7869294560334914, 1.785714801761716, 2.171083853879936, 2.989134543959161, 3.858666966566378, 4.319109148517318, 5.183196509724414, 6.105581156641971, 6.224049534205340, 6.937104404765398, 7.583678680881843, 8.471653548441130, 8.915513795267085, 9.448222347339062, 9.836015744506314, 10.36762019643803, 11.02084479207643, 11.78749265611220, 12.37076210070121, 12.81363939328449, 13.47372971928918, 13.98661233722294, 14.28105283441123, 14.72012021690504, 15.49843059955769

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