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Properties

Label 2-2800-28.23-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: $\chi_{2800} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.904220977\)
\(L(\frac12)\) \(\approx\) \(1.904220977\)
\(L(1)\) 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

−8.651746573670143923419291260491, −7.950134946399620566929923618986, −7.53758556063866031248346791656, −6.62261830364806363565868760777, −6.16309728399489559298247429638, −4.57761075023519356061787537343, −3.83804117434463970003234380751, −2.99769289181205920909227961806, −2.21416939634295556955218188583, −1.05360544942746112783773231755, 1.97198077211416861539434520807, 2.68667225747764379254401762116, 3.53740111545661300441772818831, 4.17492027875520094134526216986, 5.19746664228429477241012078262, 6.01535145275547496174781841091, 7.07303648468975443445846057408, 7.996662931193688149785203249798, 8.490224826170751356798806729318, 9.156650924377871290781679716245

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