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Properties

Label 2-800-5.4-c3-0-6
Degree $2$
Conductor $800$
Sign $-0.894 - 0.447i$
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  + 6.32i·3-s − 18.9i·7-s − 13.0·9-s + 12.6·11-s − 38i·13-s + 34i·17-s − 101.·19-s + 120.·21-s + 82.2i·23-s + 88.5i·27-s − 270·29-s + 341.·31-s + 80.0i·33-s + 206i·37-s + 240.·39-s + ⋯
L(s)  = 1  + 1.21i·3-s − 1.02i·7-s − 0.481·9-s + 0.346·11-s − 0.810i·13-s + 0.485i·17-s − 1.22·19-s + 1.24·21-s + 0.745i·23-s + 0.631i·27-s − 1.72·29-s + 1.97·31-s + 0.422i·33-s + 0.915i·37-s + 0.986·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 800,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.141933547\)
\(L(\frac12)\) \(\approx\) \(1.141933547\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.32iT - 27T^{2} \)
7 \( 1 + 18.9iT - 343T^{2} \)
11 \( 1 - 12.6T + 1.33e3T^{2} \)
13 \( 1 + 38iT - 2.19e3T^{2} \)
17 \( 1 - 34iT - 4.91e3T^{2} \)
19 \( 1 + 101.T + 6.85e3T^{2} \)
23 \( 1 - 82.2iT - 1.21e4T^{2} \)
29 \( 1 + 270T + 2.43e4T^{2} \)
31 \( 1 - 341.T + 2.97e4T^{2} \)
37 \( 1 - 206iT - 5.06e4T^{2} \)
41 \( 1 + 270T + 6.89e4T^{2} \)
43 \( 1 - 537. iT - 7.95e4T^{2} \)
47 \( 1 - 132. iT - 1.03e5T^{2} \)
53 \( 1 - 258iT - 1.48e5T^{2} \)
59 \( 1 + 75.8T + 2.05e5T^{2} \)
61 \( 1 + 250T + 2.26e5T^{2} \)
67 \( 1 - 815. iT - 3.00e5T^{2} \)
71 \( 1 - 645.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 + 278.T + 4.93e5T^{2} \)
83 \( 1 + 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 + 890T + 7.04e5T^{2} \)
97 \( 1 + 254iT - 9.12e5T^{2} \)
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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

−10.14289355959184606036299783928, −9.707471539921153495097838436682, −8.612436073646322374936549187432, −7.79064205441827701702314191749, −6.73497284007487287629271409317, −5.71873574954797315631081575722, −4.57267198290134705285973071264, −4.02417970991783168220829642503, −3.05971990108445470609390243194, −1.30148192195465917127077492268, 0.30754255808547337006942640358, 1.81875124150651599653546107399, 2.42410657128734244217791672364, 3.97356073715812962746514308243, 5.18595101925715461520212678303, 6.31859945688544738782486835655, 6.74130011157621801575001352603, 7.76977921106680296347720874053, 8.642008820311230298211754083432, 9.248729011616353206255123907402

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