Properties

Label 2-22848-1.1-c1-0-1
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 − 11-s − 13-s − 3·15-s − 17-s − 6·19-s − 21-s − 2·23-s + 4·25-s + 27-s + 2·29-s − 33-s + 3·35-s − 5·37-s − 39-s + 4·41-s + 9·43-s − 3·45-s + 49-s − 51-s − 11·53-s + 3·55-s − 6·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-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 + 0.371·29-s − 0.174·33-s + 0.507·35-s − 0.821·37-s − 0.160·39-s + 0.624·41-s + 1.37·43-s − 0.447·45-s + 1/7·49-s − 0.140·51-s − 1.51·53-s + 0.404·55-s − 0.794·57-s − 0.520·59-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

\(L(1)\) \(\approx\) \(0.7377844084\)
\(L(\frac12)\) \(\approx\) \(0.7377844084\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−15.47824663186652, −15.06326089889884, −14.40789422316260, −14.04852252254674, −13.17810637185583, −12.79029284046572, −12.24711470979065, −11.82994193856732, −10.93586412200157, −10.74209176479328, −9.962794227977560, −9.318921148887913, −8.681542755987569, −8.257780959689219, −7.652546070147754, −7.229753010999401, −6.505000907715964, −5.899253008329120, −4.905436442506254, −4.257799101913611, −3.925635545255822, −3.066442040879787, −2.531060877970850, −1.579596531718236, −0.3262582033069080, 0.3262582033069080, 1.579596531718236, 2.531060877970850, 3.066442040879787, 3.925635545255822, 4.257799101913611, 4.905436442506254, 5.899253008329120, 6.505000907715964, 7.229753010999401, 7.652546070147754, 8.257780959689219, 8.681542755987569, 9.318921148887913, 9.962794227977560, 10.74209176479328, 10.93586412200157, 11.82994193856732, 12.24711470979065, 12.79029284046572, 13.17810637185583, 14.04852252254674, 14.40789422316260, 15.06326089889884, 15.47824663186652

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