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Properties

Label 2-2448-612.319-c0-0-0
Degree $2$
Conductor $2448$
Sign $-0.0352 - 0.999i$
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  + (−0.258 − 0.965i)3-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)11-s + (−0.5 + 0.866i)13-s + i·17-s − 1.41·19-s + (−0.965 + 0.258i)23-s + (−0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (−0.366 + 1.36i)29-s + (−0.258 − 0.965i)31-s + 33-s + (0.965 + 0.258i)39-s + (−0.707 − 1.22i)43-s + (1.22 − 0.707i)47-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)11-s + (−0.5 + 0.866i)13-s + i·17-s − 1.41·19-s + (−0.965 + 0.258i)23-s + (−0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (−0.366 + 1.36i)29-s + (−0.258 − 0.965i)31-s + 33-s + (0.965 + 0.258i)39-s + (−0.707 − 1.22i)43-s + (1.22 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-0.0352 - 0.999i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ -0.0352 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4867114662\)
\(L(\frac12)\) \(\approx\) \(0.4867114662\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
17 \( 1 - iT \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.154873166082899024720407419128, −8.507793439834791300012848191063, −7.57315976397439218461208662807, −7.14973124770787250808605854248, −6.26624818435623340021558968412, −5.62089316676089411429423694500, −4.57683489714820784154967911845, −3.72811360997501213401724236575, −2.15645354812300980401736537171, −1.83955429278681031294408746122, 0.30613379476735466629106623673, 2.38399288607482362660146515957, 3.22850111984464003924754865316, 4.20382580914920377038699096915, 4.89636758743515717473305199929, 5.89394542119884411945212429051, 6.23442156154293240707367548336, 7.60764019700318698491969201225, 8.228066958023316871119754254362, 9.012490858478355251211358235723

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