↑

Properties

Label 2-2448-204.143-c0-0-0
Degree $2$
Conductor $2448$
Sign $0.328 + 0.944i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.328 + 0.944i$
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: \((2,\ 2448,\ (\ :0),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5832936152\)
\(L(\frac12)\) \(\approx\) \(0.5832936152\)
\(L(1)\) 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 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.972280190702873370130292690159, −7.88266754778308352207051786580, −7.61554105944769954551400789610, −6.91866305738548554917121236774, −5.95741356392638439225101080632, −4.66399015139548221417280328568, −4.24393106642642808087822296110, −3.31032222516902181398606246752, −2.40068756010791423141420975479, −0.45283156965622278424304080244, 1.17272242585172904155747782334, 2.89007216585672326142885119513, 3.65926023599157923409855716015, 4.45630524940959734975249685559, 5.12253100531067466792651955101, 6.26786443542922257407100991032, 7.26211939936515662853238351828, 7.75044230650823976266951682159, 8.419407757053120020520001018239, 9.011259774614765796236877112546

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