Properties

Label 2-22848-1.1-c1-0-19
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s + 11-s − 13-s − 15-s − 17-s + 19-s + 21-s + 3·23-s − 4·25-s + 27-s + 2·29-s + 33-s − 35-s + 6·37-s − 39-s − 41-s + 5·43-s − 45-s − 12·47-s + 49-s − 51-s − 55-s + 57-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.218·21-s + 0.625·23-s − 4/5·25-s + 0.192·27-s + 0.371·29-s + 0.174·33-s − 0.169·35-s + 0.986·37-s − 0.160·39-s − 0.156·41-s + 0.762·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.140·51-s − 0.134·55-s + 0.132·57-s + 0.256·61-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

\(L(1)\) \(\approx\) \(2.620023304\)
\(L(\frac12)\) \(\approx\) \(2.620023304\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.28417141346702, −15.00753041099238, −14.46411217234412, −13.91035828879110, −13.38799006354783, −12.82248092764270, −12.23633818238016, −11.61105162501048, −11.24605934426482, −10.54640112923208, −9.897378116479734, −9.338124780493756, −8.861890293734129, −8.071784764466131, −7.825297767419185, −7.121111643035421, −6.516470636171469, −5.810027740272265, −4.947202095856909, −4.477233670699155, −3.760030422190277, −3.128456663772249, −2.357818666369689, −1.595022626537393, −0.6404783163993042, 0.6404783163993042, 1.595022626537393, 2.357818666369689, 3.128456663772249, 3.760030422190277, 4.477233670699155, 4.947202095856909, 5.810027740272265, 6.516470636171469, 7.121111643035421, 7.825297767419185, 8.071784764466131, 8.861890293734129, 9.338124780493756, 9.897378116479734, 10.54640112923208, 11.24605934426482, 11.61105162501048, 12.23633818238016, 12.82248092764270, 13.38799006354783, 13.91035828879110, 14.46411217234412, 15.00753041099238, 15.28417141346702

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