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Properties

Label 2-2448-204.143-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.999 + 0.00538i$
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  + (1.92 − 0.382i)5-s + (−0.541 + 0.541i)13-s + i·17-s + (2.63 − 1.08i)25-s + (0.324 + 1.63i)29-s + (1.08 − 1.63i)37-s + (−1.08 − 0.216i)41-s + (−0.923 − 0.382i)49-s + (−0.707 − 0.292i)53-s + (−1.63 − 0.324i)61-s + (−0.834 + 1.24i)65-s + (−0.382 − 1.92i)73-s + (0.382 + 1.92i)85-s + (0.541 − 0.541i)89-s + (1.08 − 0.216i)97-s + ⋯
L(s)  = 1  + (1.92 − 0.382i)5-s + (−0.541 + 0.541i)13-s + i·17-s + (2.63 − 1.08i)25-s + (0.324 + 1.63i)29-s + (1.08 − 1.63i)37-s + (−1.08 − 0.216i)41-s + (−0.923 − 0.382i)49-s + (−0.707 − 0.292i)53-s + (−1.63 − 0.324i)61-s + (−0.834 + 1.24i)65-s + (−0.382 − 1.92i)73-s + (0.382 + 1.92i)85-s + (0.541 − 0.541i)89-s + (1.08 − 0.216i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.999 + 0.00538i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.999 + 0.00538i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.668261392\)
\(L(\frac12)\) \(\approx\) \(1.668261392\)
\(L(1)\) 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 \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
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 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
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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

−9.129732029081890544004737107451, −8.678841197161910598014350925556, −7.54861264013351767836896921779, −6.55980437702882575860074838147, −6.08602310104126069646110004502, −5.22856069132374263722647492126, −4.60847673234576695104990858784, −3.26312736417885952956850399770, −2.13940061779729048441012798485, −1.50493360291353787008980613226, 1.36325453808845905103828391990, 2.51322207374640661923634823954, 2.99282666787585403103213062234, 4.61065365357188190013537413586, 5.26951220694871638713747592808, 6.10968542375919954731786616517, 6.59167729483433782836308829048, 7.53900165000748765268401109112, 8.427352963738548716740190842972, 9.496927067737046169802575015598

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