Properties

Label 2-2800-5.4-c1-0-2
Degree $2$
Conductor $2800$
Sign $0.447 - 0.894i$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56i·3-s + i·7-s − 3.56·9-s − 2.12·11-s + 2i·13-s + 2.56i·17-s − 0.561·19-s + 2.56·21-s − 5.56i·23-s + 1.43i·27-s − 7.56·29-s + 0.876·31-s + 5.43i·33-s + 11.8i·37-s + 5.12·39-s + ⋯
L(s)  = 1  − 1.47i·3-s + 0.377i·7-s − 1.18·9-s − 0.640·11-s + 0.554i·13-s + 0.621i·17-s − 0.128·19-s + 0.558·21-s − 1.15i·23-s + 0.276i·27-s − 1.40·29-s + 0.157·31-s + 0.946i·33-s + 1.94i·37-s + 0.820·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6760218172\)
\(L(\frac12)\) \(\approx\) \(0.6760218172\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + 2.56iT - 3T^{2} \)
11 \( 1 + 2.12T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2.56iT - 17T^{2} \)
19 \( 1 + 0.561T + 19T^{2} \)
23 \( 1 + 5.56iT - 23T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 - 0.876T + 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 + 2.43iT - 43T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 - 7.12iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 - 16.1iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 6.68T + 79T^{2} \)
83 \( 1 - 13.6iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.604205074594680986055355541828, −8.155714631460318105771461544011, −7.41185800824097086576347388063, −6.64193045596532060195599824951, −6.13929410612148289682104804336, −5.23953817822211231278150775528, −4.22495129991534024334847668274, −2.94498433768572369258952784987, −2.15145876266324209318313177541, −1.27173748972588923671575047469, 0.21527709717583578563306915765, 2.06123683314587195319766088841, 3.39631878616851132415780723590, 3.70535988318055846433279057956, 4.93079388262413828629638135593, 5.21555442367481850101466335603, 6.17042539008638506195731407769, 7.38158611558906079745709669571, 7.86513560761298721556001292279, 8.987799981911364732029578975377

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