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Properties

Label 2-1440-5.3-c0-0-2
Degree $2$
Conductor $1440$
Sign $0.229 + 0.973i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
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  − i·5-s + (−1 − i)13-s + (1 − i)17-s − 25-s + (1 − i)37-s + i·49-s + (−1 − i)53-s + (−1 + i)65-s + (1 + i)73-s + (−1 − i)85-s + (1 − i)97-s + 2·101-s + 2i·109-s + (1 + i)113-s + ⋯
L(s)  = 1  − i·5-s + (−1 − i)13-s + (1 − i)17-s − 25-s + (1 − i)37-s + i·49-s + (−1 − i)53-s + (−1 + i)65-s + (1 + i)73-s + (−1 − i)85-s + (1 − i)97-s + 2·101-s + 2i·109-s + (1 + i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003579579\)
\(L(\frac12)\) \(\approx\) \(1.003579579\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.619135127605727888345857042665, −8.805170338517886325761611565369, −7.81194292611428217007354855222, −7.45672434476210826111335352555, −6.10480684256356854509112507007, −5.26430892208687826208643730303, −4.70980916418435184764549695298, −3.48938460250877756966617029190, −2.38113606625636665596598092352, −0.848296586500977528890620058934, 1.80431104230119115090236852145, 2.88015613929687152014350821595, 3.83987950428104676100348833714, 4.83510087779419007068133314720, 5.96945497089711161558334807659, 6.63557196822911011547672148525, 7.47779700011647836387520347388, 8.120543968661658975165355533118, 9.268509675723928818072588730068, 9.948291209855228656984027196396

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