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Properties

Label 2-2448-204.95-c0-0-0
Degree $2$
Conductor $2448$
Sign $0.930 - 0.366i$
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  + (0.0761 + 0.382i)5-s + (0.541 − 0.541i)13-s + i·17-s + (0.783 − 0.324i)25-s + (1.08 − 0.216i)29-s + (0.324 + 0.216i)37-s + (−0.324 + 1.63i)41-s + (0.923 + 0.382i)49-s + (−0.707 − 0.292i)53-s + (0.216 − 1.08i)61-s + (0.248 + 0.165i)65-s + (0.382 − 0.0761i)73-s + (−0.382 + 0.0761i)85-s + (−0.541 + 0.541i)89-s + (0.324 + 1.63i)97-s + ⋯
L(s)  = 1  + (0.0761 + 0.382i)5-s + (0.541 − 0.541i)13-s + i·17-s + (0.783 − 0.324i)25-s + (1.08 − 0.216i)29-s + (0.324 + 0.216i)37-s + (−0.324 + 1.63i)41-s + (0.923 + 0.382i)49-s + (−0.707 − 0.292i)53-s + (0.216 − 1.08i)61-s + (0.248 + 0.165i)65-s + (0.382 − 0.0761i)73-s + (−0.382 + 0.0761i)85-s + (−0.541 + 0.541i)89-s + (0.324 + 1.63i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.930 - 0.366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267135320\)
\(L(\frac12)\) \(\approx\) \(1.267135320\)
\(L(1)\) 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 \( 1 \)
17 \( 1 - iT \)
good5 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (-0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.324 - 1.63i)T + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
97 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
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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

−9.135532558564675132871660373875, −8.277626818536636387444873041720, −7.84293857550077912655155214176, −6.62929555387102136107234976981, −6.28424443191516359133290099538, −5.26879001866812439553929180805, −4.36523706566208486403020984502, −3.40272442510462150137650802561, −2.56618337279626266827643740496, −1.24927244131102003272740372877, 1.05349695281099571760059629402, 2.32492192217763197959778570376, 3.36234951077354721862051828994, 4.37079805424940991653867150333, 5.10281106846702713448909756374, 5.94505214079586301293447772724, 6.86766201246194826330059240133, 7.44163865910628308554350204603, 8.579737859098327287585406222993, 8.911798132145705626594208355327

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