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Properties

Label 2-2448-612.475-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.642 - 0.766i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.707 − 1.22i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 1.67i)11-s + (0.866 + 1.5i)13-s − 17-s + (−1 + 0.999i)21-s + (−0.258 − 0.448i)23-s + (−0.5 + 0.866i)25-s + (−0.707 − 0.707i)27-s + (0.965 + 1.67i)31-s + (1.36 − 1.36i)33-s + (−0.448 − 1.67i)39-s + (−0.499 − 0.866i)49-s + (0.965 + 0.258i)51-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.707 − 1.22i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 1.67i)11-s + (0.866 + 1.5i)13-s − 17-s + (−1 + 0.999i)21-s + (−0.258 − 0.448i)23-s + (−0.5 + 0.866i)25-s + (−0.707 − 0.707i)27-s + (0.965 + 1.67i)31-s + (1.36 − 1.36i)33-s + (−0.448 − 1.67i)39-s + (−0.499 − 0.866i)49-s + (0.965 + 0.258i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8026512482\)
\(L(\frac12)\) \(\approx\) \(0.8026512482\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
17 \( 1 + T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.332706971744727436968925230774, −8.324429996033810484310923645319, −7.40050811002781079619569901948, −7.00045653979472189761232917159, −6.34471045743533958357338394671, −5.07678373835831972381948208667, −4.56179941829918458348279997179, −3.98276600668608489005318044500, −2.13781875807067775442209514239, −1.38070059276525858755592470640, 0.64804666363117938097746212262, 2.27707770945956520312059423902, 3.26803400161393338296073287345, 4.39052015168320591692834817325, 5.37985141759361526999771872116, 5.83581362159299110610131049127, 6.23412650630756119100065930892, 7.70335495008791127713935543703, 8.310452916058209500337772315173, 8.827780657390144395210081429721

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