Properties

Label 2-22848-1.1-c1-0-38
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 − 21-s + 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s + 2·35-s − 10·37-s − 6·39-s − 6·41-s + 12·43-s + 2·45-s + 49-s − 51-s + 10·53-s − 8·59-s − 6·61-s + 63-s + 12·65-s + 12·67-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 − 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.338·35-s − 1.64·37-s − 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.298·45-s + 1/7·49-s − 0.140·51-s + 1.37·53-s − 1.04·59-s − 0.768·61-s + 0.125·63-s + 1.48·65-s + 1.46·67-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

\(L(1)\) \(\approx\) \(3.199112822\)
\(L(\frac12)\) \(\approx\) \(3.199112822\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.53172682501417, −15.13252490850410, −14.12529695751070, −13.87632676736283, −13.40711183862861, −12.85588560824027, −12.13928173439914, −11.70789110078413, −11.01995226712181, −10.48047167967827, −10.28557194725368, −9.303154366420942, −8.914753199336603, −8.318465029992625, −7.636839967880307, −6.736070207106995, −6.442888401671257, −5.773353069252698, −5.227515401030605, −4.654736170286014, −3.816806648141567, −3.096971238824897, −2.229869784200787, −1.321324863966388, −0.8629385220287044, 0.8629385220287044, 1.321324863966388, 2.229869784200787, 3.096971238824897, 3.816806648141567, 4.654736170286014, 5.227515401030605, 5.773353069252698, 6.442888401671257, 6.736070207106995, 7.636839967880307, 8.318465029992625, 8.914753199336603, 9.303154366420942, 10.28557194725368, 10.48047167967827, 11.01995226712181, 11.70789110078413, 12.13928173439914, 12.85588560824027, 13.40711183862861, 13.87632676736283, 14.12529695751070, 15.13252490850410, 15.53172682501417

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