Properties

Label 2-22848-1.1-c1-0-23
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 − 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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
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

−15.69502173587414, −15.32128694873723, −14.78238428673364, −14.37531187975436, −13.34348843206832, −12.97898007501918, −12.31226765970668, −12.09666415698453, −11.37640659667415, −10.91600509588319, −10.41344709381269, −9.798227761437234, −9.038508234637717, −8.523240588507460, −7.782308176376744, −7.300167487289325, −6.914509237742370, −6.041463905933278, −5.491373360876384, −4.697555761757052, −4.170009916045497, −3.626684146614978, −2.764447837632673, −1.973972016896194, −0.7005601408151902, 0, 0.7005601408151902, 1.973972016896194, 2.764447837632673, 3.626684146614978, 4.170009916045497, 4.697555761757052, 5.491373360876384, 6.041463905933278, 6.914509237742370, 7.300167487289325, 7.782308176376744, 8.523240588507460, 9.038508234637717, 9.798227761437234, 10.41344709381269, 10.91600509588319, 11.37640659667415, 12.09666415698453, 12.31226765970668, 12.97898007501918, 13.34348843206832, 14.37531187975436, 14.78238428673364, 15.32128694873723, 15.69502173587414

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