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Properties

Label 2-800-32.21-c1-0-69
Degree $2$
Conductor $800$
Sign $-0.195 - 0.980i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (0.707 − 0.292i)3-s − 2.00·4-s + (−0.414 − 1.00i)6-s + (−1 − i)7-s + 2.82i·8-s + (−1.70 + 1.70i)9-s + (−4.12 − 1.70i)11-s + (−1.41 + 0.585i)12-s + (−0.292 − 0.707i)13-s + (−1.41 + 1.41i)14-s + 4.00·16-s − 2.82i·17-s + (2.41 + 2.41i)18-s + (1.53 + 3.70i)19-s + ⋯
L(s)  = 1  − 0.999i·2-s + (0.408 − 0.169i)3-s − 1.00·4-s + (−0.169 − 0.408i)6-s + (−0.377 − 0.377i)7-s + 1.00i·8-s + (−0.569 + 0.569i)9-s + (−1.24 − 0.514i)11-s + (−0.408 + 0.169i)12-s + (−0.0812 − 0.196i)13-s + (−0.377 + 0.377i)14-s + 1.00·16-s − 0.685i·17-s + (0.569 + 0.569i)18-s + (0.352 + 0.850i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 \)
good3 \( 1 + (-0.707 + 0.292i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (4.12 + 1.70i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.292 + 0.707i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (-1.53 - 3.70i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.82 - 5.82i)T - 23iT^{2} \)
29 \( 1 + (3.12 - 1.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (0.292 - 0.707i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.171 - 0.171i)T - 41iT^{2} \)
43 \( 1 + (4.70 + 1.94i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.343iT - 47T^{2} \)
53 \( 1 + (-1.12 - 0.464i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.87 - 4.53i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.53 + 2.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.82 + 5.82i)T + 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (1.87 + 4.53i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.65 - 8.65i)T + 89iT^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−9.872646605310008168985372169941, −8.912186880040474685730745329412, −8.004550216472812858035360192627, −7.49522591645279530623130051672, −5.73555343818933855075921472411, −5.16683603362759257068352797762, −3.71471335167131132203540339668, −2.97556159879995253415990871864, −1.87417386993632887036259059602, 0, 2.48302901028086144495066347102, 3.67755908380458093349812821284, 4.75666008916035581234821039073, 5.74337484532709293090788605601, 6.47842740048033495269320342687, 7.53343111764430450896276565075, 8.292609621317742343109269376971, 9.026959375219524535402227285098, 9.770989852915791038973192794945

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