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Properties

Label 2-800-40.29-c1-0-15
Degree $2$
Conductor $800$
Sign $-0.316 + 0.948i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
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  + 3-s − 2i·7-s − 2·9-s − 5i·11-s − 6·13-s − 3i·17-s − i·19-s − 2i·21-s − 4i·23-s − 5·27-s + 6i·29-s + 8·31-s − 5i·33-s − 2·37-s − 6·39-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755i·7-s − 0.666·9-s − 1.50i·11-s − 1.66·13-s − 0.727i·17-s − 0.229i·19-s − 0.436i·21-s − 0.834i·23-s − 0.962·27-s + 1.11i·29-s + 1.43·31-s − 0.870i·33-s − 0.328·37-s − 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727197 - 1.00893i\)
\(L(\frac12)\) \(\approx\) \(0.727197 - 1.00893i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + 5T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.987142260298298385209456979569, −9.060959574021930535372260896490, −8.361243452596293051547985876716, −7.52112165838780111651625021751, −6.65600947822911606397131805922, −5.50996855221249403442753780401, −4.55334649367177553981423034236, −3.26616880810829975877973639717, −2.54877433710361515044148919038, −0.54159427436150517661169745230, 2.07440600972409045506694274946, 2.74301250608414464673336532883, 4.18531602986919066814438565275, 5.14418779717779421825918832717, 6.10538948341972315551104527552, 7.30182299322499789568555367799, 7.931617904026646086757758050846, 8.880629356156267204066230190914, 9.698213947827877298013482403159, 10.17370566065063935513034620934

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