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Properties

Label 2-800-5.4-c3-0-7
Degree $2$
Conductor $800$
Sign $-0.447 + 0.894i$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.19i·3-s + 21.7i·7-s − 40.0·9-s − 37.8·11-s + 52.7i·13-s + 99.9i·17-s + 116.·19-s − 177.·21-s − 37.7i·23-s − 107. i·27-s − 218.·29-s + 67.3·31-s − 310. i·33-s + 362. i·37-s − 431.·39-s + ⋯
L(s)  = 1  + 1.57i·3-s + 1.17i·7-s − 1.48·9-s − 1.03·11-s + 1.12i·13-s + 1.42i·17-s + 1.40·19-s − 1.84·21-s − 0.342i·23-s − 0.764i·27-s − 1.40·29-s + 0.390·31-s − 1.63i·33-s + 1.61i·37-s − 1.77·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.181151185\)
\(L(\frac12)\) \(\approx\) \(1.181151185\)
\(L(\frac{5}{2})\) 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 \( 1 - 8.19iT - 27T^{2} \)
7 \( 1 - 21.7iT - 343T^{2} \)
11 \( 1 + 37.8T + 1.33e3T^{2} \)
13 \( 1 - 52.7iT - 2.19e3T^{2} \)
17 \( 1 - 99.9iT - 4.91e3T^{2} \)
19 \( 1 - 116.T + 6.85e3T^{2} \)
23 \( 1 + 37.7iT - 1.21e4T^{2} \)
29 \( 1 + 218.T + 2.43e4T^{2} \)
31 \( 1 - 67.3T + 2.97e4T^{2} \)
37 \( 1 - 362. iT - 5.06e4T^{2} \)
41 \( 1 + 291.T + 6.89e4T^{2} \)
43 \( 1 + 183. iT - 7.95e4T^{2} \)
47 \( 1 + 443. iT - 1.03e5T^{2} \)
53 \( 1 + 416. iT - 1.48e5T^{2} \)
59 \( 1 - 828.T + 2.05e5T^{2} \)
61 \( 1 - 442.T + 2.26e5T^{2} \)
67 \( 1 - 570. iT - 3.00e5T^{2} \)
71 \( 1 + 341.T + 3.57e5T^{2} \)
73 \( 1 - 506. iT - 3.89e5T^{2} \)
79 \( 1 - 426.T + 4.93e5T^{2} \)
83 \( 1 - 982. iT - 5.71e5T^{2} \)
89 \( 1 + 926.T + 7.04e5T^{2} \)
97 \( 1 + 1.88e3iT - 9.12e5T^{2} \)
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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

−10.04710487273998710065948439040, −9.913090538300906763288610807391, −8.736209475016375841112912285210, −8.383217586377164049076655290021, −6.91487137359977685954729814118, −5.59576697277347327607754188392, −5.23191700732941212283306746644, −4.10733248566027376108221923412, −3.21264591883700515587046527044, −2.02615698110263831086645565274, 0.35060287253539888030017114947, 1.06733013463849606922839094729, 2.46843800263408243349692115452, 3.42378774097888648338711154720, 5.05093189709394774470216649072, 5.81493696725411369563310169498, 7.06985324323973303966532351864, 7.56227078498362959569574422748, 7.892934063024697504855487458449, 9.278095624516661068907089401215

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