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Properties

Label 2-800-200.109-c1-0-7
Degree $2$
Conductor $800$
Sign $-0.996 - 0.0831i$
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  + (0.970 + 2.98i)3-s + (−0.762 + 2.10i)5-s + 1.71i·7-s + (−5.55 + 4.03i)9-s + (−1.35 + 1.86i)11-s + (4.80 − 3.49i)13-s + (−7.01 − 0.236i)15-s + (−2.53 − 0.824i)17-s + (5.48 + 1.78i)19-s + (−5.13 + 1.66i)21-s + (0.393 − 0.542i)23-s + (−3.83 − 3.20i)25-s + (−9.82 − 7.13i)27-s + (−1.22 + 0.398i)29-s + (−0.362 + 1.11i)31-s + ⋯
L(s)  = 1  + (0.560 + 1.72i)3-s + (−0.340 + 0.940i)5-s + 0.649i·7-s + (−1.85 + 1.34i)9-s + (−0.407 + 0.561i)11-s + (1.33 − 0.968i)13-s + (−1.81 − 0.0609i)15-s + (−0.615 − 0.199i)17-s + (1.25 + 0.408i)19-s + (−1.11 + 0.363i)21-s + (0.0821 − 0.113i)23-s + (−0.767 − 0.640i)25-s + (−1.89 − 1.37i)27-s + (−0.227 + 0.0739i)29-s + (−0.0651 + 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.996 - 0.0831i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.996 - 0.0831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0661748 + 1.58835i\)
\(L(\frac12)\) \(\approx\) \(0.0661748 + 1.58835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.762 - 2.10i)T \)
good3 \( 1 + (-0.970 - 2.98i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 1.71iT - 7T^{2} \)
11 \( 1 + (1.35 - 1.86i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.80 + 3.49i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.53 + 0.824i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.48 - 1.78i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.393 + 0.542i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.22 - 0.398i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.362 - 1.11i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.95 + 2.14i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.21 + 4.51i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 + (8.51 - 2.76i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.09 - 3.35i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.58 + 4.92i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.542 + 0.746i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.74 - 14.5i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.92 + 5.93i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.14 + 4.32i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.125 - 0.386i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.356 + 1.09i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.34 - 2.43i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (14.4 - 4.69i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.72088056974758021722734571612, −9.837198621871462349769607699338, −9.155197994402397583765178649453, −8.285027393249345015749517515989, −7.50620088652724345394062288116, −6.03236610829866388555978645415, −5.27034169672746804854526800706, −4.13734556123153022405740220985, −3.31649533113441700706053358549, −2.56957460401325407962129763805, 0.77793741543710038976124493496, 1.66474923993038348660152099036, 3.12347379800184368298944827940, 4.23335552957133670979393298379, 5.66913076560616279416226448239, 6.53921790946287435101610084939, 7.39969691081829284944904002964, 8.089819448676689174318257030813, 8.748801114165235289011215597355, 9.460202098260934609281591431001

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