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Properties

Label 2-2448-204.95-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.655 + 0.755i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.655 + 0.755i$
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: \((2,\ 2448,\ (\ :0),\ 0.655 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175651090\)
\(L(\frac12)\) \(\approx\) \(1.175651090\)
\(L(1)\) 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

−9.024895591754045374499524753078, −8.318363319300296040750954122657, −7.49451836387155711018144514868, −6.80911354146665333120182917702, −5.77481248380799014049409902671, −5.16598491164579230394642012498, −4.21811004836707826202413236547, −3.30983993524438352421558164117, −2.27239757855052327527199304874, −0.876002160616828662597718224536, 1.41147370603204981901870197182, 2.55403290314821654775400657500, 3.62489327818235690602422743247, 4.30067448633473688714225753827, 5.41540351764521802988118224644, 6.19053882790449562030478086010, 6.89441400110775264924994077248, 7.69775327578153442494774693423, 8.499404599033946670747189160918, 9.168348169265850286749204615231

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