Properties

Label 2-22848-1.1-c1-0-11
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 − 4·11-s − 6·13-s + 2·15-s + 17-s − 21-s − 25-s + 27-s − 2·29-s − 4·33-s − 2·35-s − 6·37-s − 6·39-s + 2·41-s + 4·43-s + 2·45-s + 49-s + 51-s + 2·53-s − 8·55-s − 4·59-s − 2·61-s − 63-s − 12·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.242·17-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.338·35-s − 0.986·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s + 0.274·53-s − 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.125·63-s − 1.48·65-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

\(L(1)\) \(\approx\) \(2.042795542\)
\(L(\frac12)\) \(\approx\) \(2.042795542\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.55842892647441, −14.87554124354921, −14.35151493862010, −13.87432651422426, −13.39586113103482, −12.77422655780827, −12.46111782897887, −11.82311121225284, −10.91907659924361, −10.38332739990192, −9.861993480681204, −9.573012309039249, −8.931023347121074, −8.209867696203898, −7.520458611917635, −7.257778597335531, −6.396196111815289, −5.706451131450707, −5.153127228839178, −4.626794292803177, −3.650713900956539, −2.919067547839824, −2.321379102050510, −1.864889881280234, −0.5193902103930911, 0.5193902103930911, 1.864889881280234, 2.321379102050510, 2.919067547839824, 3.650713900956539, 4.626794292803177, 5.153127228839178, 5.706451131450707, 6.396196111815289, 7.257778597335531, 7.520458611917635, 8.209867696203898, 8.931023347121074, 9.573012309039249, 9.861993480681204, 10.38332739990192, 10.91907659924361, 11.82311121225284, 12.46111782897887, 12.77422655780827, 13.39586113103482, 13.87432651422426, 14.35151493862010, 14.87554124354921, 15.55842892647441

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