Properties

Label 2-22848-1.1-c1-0-4
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 + 3·11-s + 13-s + 15-s + 17-s − 6·19-s + 21-s + 6·23-s − 4·25-s − 27-s − 2·29-s − 4·31-s − 3·33-s + 35-s − 9·37-s − 39-s − 7·43-s − 45-s + 12·47-s + 49-s − 51-s + 13·53-s − 3·55-s + 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-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 − 0.371·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s − 1.47·37-s − 0.160·39-s − 1.06·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.140·51-s + 1.78·53-s − 0.404·55-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: \(0\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112509367\)
\(L(\frac12)\) \(\approx\) \(1.112509367\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−15.48667229591338, −14.95167187423055, −14.64984501319325, −13.60061911389090, −13.46391835269156, −12.62717100692548, −12.12910179091944, −11.84638083853110, −10.95041459179831, −10.78094081682521, −10.07992320353585, −9.335705372676097, −8.846206948529134, −8.363155717740219, −7.404398958291202, −7.030749275769652, −6.431363986714562, −5.815853732397044, −5.217633064860817, −4.377058248769697, −3.850091465581208, −3.309048717409457, −2.223652183289430, −1.437706594115997, −0.4547926342825364, 0.4547926342825364, 1.437706594115997, 2.223652183289430, 3.309048717409457, 3.850091465581208, 4.377058248769697, 5.217633064860817, 5.815853732397044, 6.431363986714562, 7.030749275769652, 7.404398958291202, 8.363155717740219, 8.846206948529134, 9.335705372676097, 10.07992320353585, 10.78094081682521, 10.95041459179831, 11.84638083853110, 12.12910179091944, 12.62717100692548, 13.46391835269156, 13.60061911389090, 14.64984501319325, 14.95167187423055, 15.48667229591338

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