Properties

Label 2-22848-1.1-c1-0-39
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 + 5·11-s + 5·13-s − 15-s + 17-s + 5·19-s + 21-s − 23-s − 4·25-s + 27-s + 6·29-s − 6·31-s + 5·33-s − 35-s − 4·37-s + 5·39-s + 7·41-s + 7·43-s − 45-s + 6·47-s + 49-s + 51-s − 6·53-s − 5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s − 0.258·15-s + 0.242·17-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.870·33-s − 0.169·35-s − 0.657·37-s + 0.800·39-s + 1.09·41-s + 1.06·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 0.674·55-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

\(L(1)\) \(\approx\) \(3.933810803\)
\(L(\frac12)\) \(\approx\) \(3.933810803\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.63750108429212, −14.87793771777711, −14.24530545602002, −13.97048793461462, −13.62148386418932, −12.65174845808692, −12.24078519710248, −11.69537420720468, −11.09619763038043, −10.74355336122248, −9.774871255921635, −9.338981389014914, −8.814101068415306, −8.310405871022349, −7.587566011841973, −7.264443952672789, −6.275491630751878, −6.010379775544353, −5.041714373569031, −4.283605513654096, −3.652733260020100, −3.408526372910776, −2.284575080986233, −1.418361312704282, −0.8775937588071930, 0.8775937588071930, 1.418361312704282, 2.284575080986233, 3.408526372910776, 3.652733260020100, 4.283605513654096, 5.041714373569031, 6.010379775544353, 6.275491630751878, 7.264443952672789, 7.587566011841973, 8.310405871022349, 8.814101068415306, 9.338981389014914, 9.774871255921635, 10.74355336122248, 11.09619763038043, 11.69537420720468, 12.24078519710248, 12.65174845808692, 13.62148386418932, 13.97048793461462, 14.24530545602002, 14.87793771777711, 15.63750108429212

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