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Properties

Label 2-2800-112.13-c0-0-2
Degree $2$
Conductor $2800$
Sign $0.130 - 0.991i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s + i·9-s + (−0.366 + 0.366i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.133i)22-s + 1.73i·23-s + (0.866 − 0.499i)28-s + (1.36 + 1.36i)29-s + (−0.866 + 0.499i)32-s + (−0.866 + 0.499i)36-s + (1.36 − 1.36i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s + i·9-s + (−0.366 + 0.366i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.133i)22-s + 1.73i·23-s + (0.866 − 0.499i)28-s + (1.36 + 1.36i)29-s + (−0.866 + 0.499i)32-s + (−0.866 + 0.499i)36-s + (1.36 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.989677663\)
\(L(\frac12)\) \(\approx\) \(1.989677663\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
11 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
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

−8.988764002977549724385397945846, −8.046428366948525250654177460452, −7.46507894828357390024682317877, −7.05150330354819897548032764474, −5.99699205000288326525632930047, −5.18901283686349544914990686685, −4.57525383404448542329885554867, −3.73615586618674469339491882337, −2.81449298263378376416571142765, −1.68554424073358877335743557604, 1.00817359630425136200128940074, 2.57870666444971023486522729110, 2.86507838410438389527075362615, 4.15773104051092303165787506773, 4.72991320994557283054988407035, 5.93749467340088512084600604449, 6.11566569591925571779069975154, 7.00473474712929042127019522326, 8.232152750179858355471482602610, 8.826699893317861364811193468704

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