Properties

Label 2-800-160.109-c1-0-50
Degree $2$
Conductor $800$
Sign $0.964 - 0.264i$
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  + 1.41·2-s + (1.70 + 0.707i)3-s + 2.00·4-s + (2.41 + 1.00i)6-s + (−1 − i)7-s + 2.82·8-s + (0.292 + 0.292i)9-s + (0.121 + 0.292i)11-s + (3.41 + 1.41i)12-s + (−0.707 + 1.70i)13-s + (−1.41 − 1.41i)14-s + 4.00·16-s + 2.82·17-s + (0.414 + 0.414i)18-s + (5.53 + 2.29i)19-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.985 + 0.408i)3-s + 1.00·4-s + (0.985 + 0.408i)6-s + (−0.377 − 0.377i)7-s + 1.00·8-s + (0.0976 + 0.0976i)9-s + (0.0365 + 0.0883i)11-s + (0.985 + 0.408i)12-s + (−0.196 + 0.473i)13-s + (−0.377 − 0.377i)14-s + 1.00·16-s + 0.685·17-s + (0.0976 + 0.0976i)18-s + (1.26 + 0.526i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.964 - 0.264i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.81495 + 0.512922i\)
\(L(\frac12)\) \(\approx\) \(3.81495 + 0.512922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 \)
good3 \( 1 + (-1.70 - 0.707i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (-0.121 - 0.292i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.707 - 1.70i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (-5.53 - 2.29i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.171 - 0.171i)T - 23iT^{2} \)
29 \( 1 + (1.12 - 2.70i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (0.707 + 1.70i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.82 + 5.82i)T + 41iT^{2} \)
43 \( 1 + (7.94 - 3.29i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + (7.53 - 3.12i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-6.12 + 2.53i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.70 + 1.53i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (0.171 - 0.171i)T - 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (2.53 - 6.12i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.65 + 2.65i)T - 89iT^{2} \)
97 \( 1 - 1.51iT - 97T^{2} \)
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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

−10.10529672011273490397141688628, −9.686800952322329674013335442098, −8.543276096202658816834926319789, −7.60936382454298181552023184775, −6.84943482000778471275407284590, −5.74208235250230911634150575838, −4.77355466151576468512753529082, −3.55412192078164351828073216671, −3.25167052751125701823793129780, −1.79253969719500689894301592972, 1.70344967858267325742162186253, 2.97023342899124050981092404145, 3.37108579479320447484646415189, 4.90721655188281905966114444669, 5.69013060739601956222802935594, 6.77439069143860901900795580709, 7.62972637982715600955421154396, 8.270125356094303007889719794781, 9.406574591669324426454609371990, 10.19575804478822415300372202474

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