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Properties

Label 2-800-32.5-c1-0-20
Degree $2$
Conductor $800$
Sign $-0.124 + 0.992i$
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  + (−0.349 − 1.37i)2-s + (−1.06 − 2.56i)3-s + (−1.75 + 0.958i)4-s + (−3.13 + 2.35i)6-s + (2.94 + 2.94i)7-s + (1.92 + 2.06i)8-s + (−3.31 + 3.31i)9-s + (−0.569 + 1.37i)11-s + (4.31 + 3.47i)12-s + (4.53 − 1.87i)13-s + (3.00 − 5.05i)14-s + (2.16 − 3.36i)16-s + 2.78i·17-s + (5.70 + 3.38i)18-s + (2.08 − 0.863i)19-s + ⋯
L(s)  = 1  + (−0.247 − 0.968i)2-s + (−0.612 − 1.47i)3-s + (−0.877 + 0.479i)4-s + (−1.28 + 0.959i)6-s + (1.11 + 1.11i)7-s + (0.681 + 0.731i)8-s + (−1.10 + 1.10i)9-s + (−0.171 + 0.414i)11-s + (1.24 + 1.00i)12-s + (1.25 − 0.520i)13-s + (0.802 − 1.35i)14-s + (0.540 − 0.841i)16-s + 0.676i·17-s + (1.34 + 0.797i)18-s + (0.478 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781075 - 0.885094i\)
\(L(\frac12)\) \(\approx\) \(0.781075 - 0.885094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 + 1.37i)T \)
5 \( 1 \)
good3 \( 1 + (1.06 + 2.56i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-2.94 - 2.94i)T + 7iT^{2} \)
11 \( 1 + (0.569 - 1.37i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.53 + 1.87i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.78iT - 17T^{2} \)
19 \( 1 + (-2.08 + 0.863i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.26 + 3.26i)T - 23iT^{2} \)
29 \( 1 + (-2.85 - 6.89i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 0.800T + 31T^{2} \)
37 \( 1 + (-8.52 - 3.53i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.87 - 8.87i)T - 41iT^{2} \)
43 \( 1 + (-1.93 + 4.67i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.86iT - 47T^{2} \)
53 \( 1 + (2.24 - 5.42i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-13.6 - 5.64i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.72 + 11.4i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.23 + 12.6i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-2.90 - 2.90i)T + 71iT^{2} \)
73 \( 1 + (-1.13 + 1.13i)T - 73iT^{2} \)
79 \( 1 + 6.06iT - 79T^{2} \)
83 \( 1 + (-0.292 + 0.121i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.97 + 8.97i)T + 89iT^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−10.37935762032387015955773000387, −8.964507345737911039679035463767, −8.330937040572643845319906362944, −7.75230382184426793539870229858, −6.53748090233472844737522173085, −5.55237866297625741804350501631, −4.76198959905481947970648794411, −3.05246373235139164880651644540, −1.91104158030700221301092764649, −1.12430048860218761062288209216, 0.957590768184917877469261349310, 3.72651919836790531406011237684, 4.33146106470824808504866467052, 5.16594077043202959283487528857, 5.90564990359211335852318068615, 7.04436666685748596732128473705, 7.986195733534057667341609517061, 8.821804346500309062827733163990, 9.680167558354093339403733768735, 10.37090832453702238003526387124

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