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Properties

Label 2-2800-140.87-c0-0-0
Degree $2$
Conductor $2800$
Sign $0.635 - 0.772i$
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  + (0.258 + 0.965i)3-s + (0.965 − 0.258i)7-s + (0.499 + 0.866i)21-s + (−0.448 + 1.67i)23-s + (0.707 + 0.707i)27-s − i·29-s − 1.73i·41-s + (1.22 + 1.22i)43-s + (0.517 − 1.93i)47-s + (0.866 − 0.499i)49-s + (−1.5 + 0.866i)61-s + (0.448 + 1.67i)67-s − 1.73·69-s + (−0.5 + 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.965 − 0.258i)7-s + (0.499 + 0.866i)21-s + (−0.448 + 1.67i)23-s + (0.707 + 0.707i)27-s − i·29-s − 1.73i·41-s + (1.22 + 1.22i)43-s + (0.517 − 1.93i)47-s + (0.866 − 0.499i)49-s + (−1.5 + 0.866i)61-s + (0.448 + 1.67i)67-s − 1.73·69-s + (−0.5 + 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.635 - 0.772i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.635 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.558210349\)
\(L(\frac12)\) \(\approx\) \(1.558210349\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
good3 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.178243776062057348222620648328, −8.419624293261658979913237004643, −7.63846938574819939646424794373, −7.00343498669936011519754475426, −5.75111251967067739605739591204, −5.17921216576449668593841352361, −4.13422008930274537316903586797, −3.85121341714788416968025490480, −2.56610049822397251539462918392, −1.38317906057077061066604541609, 1.18464055430543940440911333060, 2.08909036094326695565958874313, 2.91010626346961249066253610873, 4.31325471089661432927955239629, 4.88870279691763862114162989176, 6.00232182082467140246700082149, 6.64438035876151753564487796336, 7.55444719655171024481453139456, 7.987816711761791565452420416789, 8.686873325503200254783891277918

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