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Properties

Label 2-2800-140.3-c0-0-1
Degree $2$
Conductor $2800$
Sign $-0.913 + 0.406i$
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  + (−0.965 + 0.258i)3-s + (−0.258 − 0.965i)7-s + (0.499 + 0.866i)21-s + (−1.67 − 0.448i)23-s + (0.707 − 0.707i)27-s + i·29-s − 1.73i·41-s + (−1.22 + 1.22i)43-s + (−1.93 − 0.517i)47-s + (−0.866 + 0.499i)49-s + (−1.5 + 0.866i)61-s + (1.67 − 0.448i)67-s + 1.73·69-s + (−0.5 + 0.866i)81-s + (−0.707 − 0.707i)83-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.258 − 0.965i)7-s + (0.499 + 0.866i)21-s + (−1.67 − 0.448i)23-s + (0.707 − 0.707i)27-s + i·29-s − 1.73i·41-s + (−1.22 + 1.22i)43-s + (−1.93 − 0.517i)47-s + (−0.866 + 0.499i)49-s + (−1.5 + 0.866i)61-s + (1.67 − 0.448i)67-s + 1.73·69-s + (−0.5 + 0.866i)81-s + (−0.707 − 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.913 + 0.406i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ -0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1772646838\)
\(L(\frac12)\) \(\approx\) \(0.1772646838\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
47 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.555682825043434341065629982002, −7.909933016773763273612973130501, −6.95148737939271005623170774420, −6.37044153380135184499876107473, −5.57695259276616464068694764786, −4.76986377542103903737347182659, −4.03502749255083192620555620997, −3.07801434572814960045173646423, −1.66545467950367897119718600871, −0.12397949692711467675813791761, 1.62314763491430904832248639597, 2.72777897458611069905323432441, 3.75511950981970946457675600626, 4.90543587653898921732795128002, 5.56686087242122368368649705949, 6.27488673035579304504923805438, 6.68488338752202175367194122599, 7.963249739230440929338888395274, 8.377302531500393066188863877981, 9.519862636694061986886542567427

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