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Properties

Label 2-2448-612.463-c0-0-0
Degree $2$
Conductor $2448$
Sign $0.978 + 0.208i$
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 + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)11-s + (−0.5 + 0.866i)13-s − i·17-s + 1.41·19-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (1.36 + 0.366i)29-s + (−0.965 + 0.258i)31-s + 33-s + (0.258 − 0.965i)39-s + (0.707 + 1.22i)43-s + (1.22 − 0.707i)47-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)11-s + (−0.5 + 0.866i)13-s − i·17-s + 1.41·19-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (1.36 + 0.366i)29-s + (−0.965 + 0.258i)31-s + 33-s + (0.258 − 0.965i)39-s + (0.707 + 1.22i)43-s + (1.22 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8128956385\)
\(L(\frac12)\) \(\approx\) \(0.8128956385\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−9.304471787642970663066894374755, −8.337831814844467370195083288319, −7.34405879029662679433337117771, −6.82494133990361003265718649957, −5.94370664835513427444457732767, −4.94319564592731051683698968903, −4.75505436204266786402187501887, −3.41245154157431612886967947151, −2.38941739711614128601662096520, −0.802871719656149056769357132621, 1.02285966557474855996019303882, 2.33758309272302245696104912706, 3.45754473282869092802615225927, 4.60469754687277721744096578527, 5.46457217265679942613953614621, 5.74941175843886985802562956542, 6.96536348569267888209071552744, 7.53105822977160166683313481130, 8.132920121995606832710786398258, 9.285343174420618359491951760232

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