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Properties

Label 2-2240-280.269-c0-0-1
Degree $2$
Conductor $2240$
Sign $0.990 + 0.134i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (0.866 − 0.5i)5-s + 7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + i·13-s + (0.866 + 1.5i)19-s + (−1.5 + 0.866i)23-s + (0.499 − 0.866i)25-s + (0.866 − 0.5i)35-s + (−0.866 − 1.5i)37-s − 1.73i·41-s + (−0.866 − 0.499i)45-s + (−0.5 − 0.866i)47-s + 49-s + (−0.866 + 1.5i)53-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + 7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + i·13-s + (0.866 + 1.5i)19-s + (−1.5 + 0.866i)23-s + (0.499 − 0.866i)25-s + (0.866 − 0.5i)35-s + (−0.866 − 1.5i)37-s − 1.73i·41-s + (−0.866 − 0.499i)45-s + (−0.5 − 0.866i)47-s + 49-s + (−0.866 + 1.5i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.990 + 0.134i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.990 + 0.134i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.548297194\)
\(L(\frac12)\) \(\approx\) \(1.548297194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.226320376975776456035163044884, −8.614299754910926722240677182850, −7.70040087453859268963695661340, −6.84297296576662972500966326592, −5.86744517432987151356022309701, −5.47603815311177803753806004610, −4.27613778820977765857500770127, −3.67166353073081897417485060296, −2.01160781365538711646454930975, −1.47474289961787118805125638376, 1.37991110394998693457353402819, 2.47115811101230923461186272652, 3.24775631497494015002957195955, 4.68077519577664867097582200480, 5.23371754083406237692600212244, 6.10751795955757101429945126832, 6.80460114202758511844855797050, 7.924357628347746508033994546960, 8.305676559851873130439070793559, 9.250271566099708563511261273176

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