Properties

Label 2-2800-112.13-c0-0-4
Degree $2$
Conductor $2800$
Sign $0.793 + 0.608i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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)2-s + (0.499 − 0.866i)4-s i·7-s + 0.999i·8-s + i·9-s + (1.36 − 1.36i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 1.86i)22-s − 1.73i·23-s + (−0.866 − 0.499i)28-s + (−0.366 − 0.366i)29-s + (0.866 + 0.499i)32-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s i·7-s + 0.999i·8-s + i·9-s + (1.36 − 1.36i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 1.86i)22-s − 1.73i·23-s + (−0.866 − 0.499i)28-s + (−0.366 − 0.366i)29-s + (0.866 + 0.499i)32-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8333887919\)
\(L(\frac12)\) \(\approx\) \(0.8333887919\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
11 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + 1.73iT - T^{2} \)
29 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
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

−8.580437093427732412155654898049, −8.357965508909631835516973292634, −7.46230189121604055678046263217, −6.63221714762607734721407553622, −6.21061330075575895872582058406, −5.12328174492143650455433284701, −4.28083942749858679936669280304, −3.18076547897161398174192070245, −1.88408178851038344834700896323, −0.77195613501369203812415190822, 1.39548302130376866648924344970, 2.15958436808587479820396933396, 3.41571398030758557080556803499, 3.96849873453520379295202759791, 5.22847631866372422830348488825, 6.26686877433515050027682524246, 6.93647851665496142405925755543, 7.54975804021047388505208163311, 8.674924498746235911296829225562, 9.156303259008504267990847613084

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