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Properties

Label 2-2800-140.3-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.722860511\)
\(L(\frac12)\) \(\approx\) \(1.722860511\)
\(L(1)\) 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

−8.997777132755997668481002201250, −8.497842400215981365366284479735, −7.48001622038481388118299704289, −7.09824150795540617408523628213, −5.77288071120696377292004877009, −5.34169786353798013305167121213, −4.20639360185063602600009130441, −3.10659646864029405254091191459, −2.55191653815412741442313300462, −1.52792713531765824169257424322, 1.10203992905035005475553564575, 2.51059743801280668261095793429, 3.22551199204039910962625427143, 4.16366215227841130762910255184, 4.75916743458038969820420467522, 5.94337633226840068576841670298, 6.77240680741726575420015408301, 7.68812502387994053910449644237, 8.062827208142646766328750934592, 9.087180501287817927694705208982

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