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Properties

Label 2-2800-28.11-c0-0-1
Degree $2$
Conductor $2800$
Sign $-0.0633 + 0.997i$
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  + (−1.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 0.866i)21-s + (1.5 − 0.866i)23-s − 1.73i·27-s + 29-s + 41-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−0.5 − 0.866i)61-s + 2·63-s + (−1.5 − 0.866i)67-s − 3·69-s + (−0.5 + 0.866i)81-s + ⋯
L(s)  = 1  + (−1.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 0.866i)21-s + (1.5 − 0.866i)23-s − 1.73i·27-s + 29-s + 41-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−0.5 − 0.866i)61-s + 2·63-s + (−1.5 − 0.866i)67-s − 3·69-s + (−0.5 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (1551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7673478814\)
\(L(\frac12)\) \(\approx\) \(0.7673478814\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.611123056329511916265594453560, −7.73785050146857986245572669526, −7.17721248035723204570324879513, −6.50110898411668536657839494563, −5.88400376381313143373700712526, −4.79136860493977764716912406003, −4.54292068620481471003078314995, −2.99488602497979532350329679122, −1.61214600715953848539190499616, −0.73844303184074498991824546866, 1.17385083407208416807156335728, 2.68804979016552641571151309832, 3.86294826901261799132690156423, 4.74784483198053718903890409375, 5.31373069866319301054837656583, 5.87129557307389728051518226024, 6.69800215487901012198829299713, 7.53532748526831033194098197811, 8.696688999095170205042108416402, 9.220014531021564237566669847883

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