Properties

Label 2-22848-1.1-c1-0-52
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 − 5-s + 7-s + 9-s − 5·11-s + 13-s − 15-s − 17-s + 6·19-s + 21-s + 6·23-s − 4·25-s + 27-s − 6·29-s + 4·31-s − 5·33-s − 35-s − 11·37-s + 39-s + 9·43-s − 45-s + 4·47-s + 49-s − 51-s + 7·53-s + 5·55-s + 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.870·33-s − 0.169·35-s − 1.80·37-s + 0.160·39-s + 1.37·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.961·53-s + 0.674·55-s + 0.794·57-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: \(1\)
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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−15.65286155076946, −15.32708154379655, −14.80353164345462, −14.00584031858243, −13.53943216148552, −13.31727900519861, −12.44691848269777, −12.05408147075165, −11.34649904139517, −10.74406532710395, −10.41971672497209, −9.603446503038976, −9.035581947493568, −8.559770138823272, −7.717961966303078, −7.570903300110984, −7.030475943868501, −5.950869443003913, −5.398178212370624, −4.829215366223101, −4.111159662181360, −3.298970912434088, −2.847608513531607, −2.013264809517635, −1.124615263500348, 0, 1.124615263500348, 2.013264809517635, 2.847608513531607, 3.298970912434088, 4.111159662181360, 4.829215366223101, 5.398178212370624, 5.950869443003913, 7.030475943868501, 7.570903300110984, 7.717961966303078, 8.559770138823272, 9.035581947493568, 9.603446503038976, 10.41971672497209, 10.74406532710395, 11.34649904139517, 12.05408147075165, 12.44691848269777, 13.31727900519861, 13.53943216148552, 14.00584031858243, 14.80353164345462, 15.32708154379655, 15.65286155076946

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