Properties

Label 2-22848-1.1-c1-0-18
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 − 13-s − 15-s − 17-s − 2·19-s − 21-s + 6·23-s − 4·25-s − 27-s − 2·29-s − 8·31-s − 5·33-s + 35-s − 37-s + 39-s + 8·41-s − 43-s + 45-s − 4·47-s + 49-s + 51-s + 5·53-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 − 0.277·13-s − 0.258·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.870·33-s + 0.169·35-s − 0.164·37-s + 0.160·39-s + 1.24·41-s − 0.152·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.686·53-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

\(L(1)\) \(\approx\) \(2.163134451\)
\(L(\frac12)\) \(\approx\) \(2.163134451\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
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

−15.41982535248155, −14.88735632460617, −14.40839359412576, −14.01944747706540, −13.14101139714156, −12.86231150248285, −12.17583575002806, −11.58488480597791, −11.16665847369365, −10.73226325646787, −9.883787749177312, −9.453908695366729, −8.912614624564931, −8.372993734905730, −7.298525320787472, −7.139301484790253, −6.314613837004857, −5.826044670434496, −5.228246800358497, −4.448184659385315, −3.971591360425034, −3.153671885316545, −2.072185010311945, −1.563918665568414, −0.6307717936590494, 0.6307717936590494, 1.563918665568414, 2.072185010311945, 3.153671885316545, 3.971591360425034, 4.448184659385315, 5.228246800358497, 5.826044670434496, 6.314613837004857, 7.139301484790253, 7.298525320787472, 8.372993734905730, 8.912614624564931, 9.453908695366729, 9.883787749177312, 10.73226325646787, 11.16665847369365, 11.58488480597791, 12.17583575002806, 12.86231150248285, 13.14101139714156, 14.01944747706540, 14.40839359412576, 14.88735632460617, 15.41982535248155

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