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Properties

Label 2-2448-17.16-c1-0-12
Degree $2$
Conductor $2448$
Sign $-0.242 - 0.970i$
Analytic cond. $19.5473$
Root an. cond. $4.42124$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·7-s + 4i·11-s + 2·13-s + (−1 − 4i)17-s + 4·19-s − 4i·23-s + 5·25-s + 4i·31-s + 8i·37-s + 8i·41-s + 4·43-s − 8·47-s − 9·49-s − 6·53-s − 12·59-s + ⋯
L(s)  = 1  + 1.51i·7-s + 1.20i·11-s + 0.554·13-s + (−0.242 − 0.970i)17-s + 0.917·19-s − 0.834i·23-s + 25-s + 0.718i·31-s + 1.31i·37-s + 1.24i·41-s + 0.609·43-s − 1.16·47-s − 1.28·49-s − 0.824·53-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(19.5473\)
Root analytic conductor: \(4.42124\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :1/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.607872956\)
\(L(\frac12)\) \(\approx\) \(1.607872956\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + (1 + 4i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−9.225287709285959191736619465834, −8.478535596709911215007179621237, −7.70263282682119755145400937233, −6.73094065576395437782977227829, −6.16631158193617952287891358636, −4.96710679513710451000328063963, −4.78669127463215302246567304886, −3.19891141047355361785014440707, −2.55317958112811488980896090028, −1.41278891630442857583352756514, 0.57352570508392388865017562551, 1.57571465254573096698888981131, 3.21488755984848636972305576028, 3.73200549467605985910475138340, 4.62365877530828677142891695210, 5.73637868312127189236441813779, 6.33860563095576610389997167655, 7.36375332958445321348303956078, 7.78987309809769759559479969485, 8.754876984127810087531641499762

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