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Properties

Label 2-800-40.27-c1-0-13
Degree $2$
Conductor $800$
Sign $0.229 + 0.973i$
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  + (2 − 2i)3-s − 5i·9-s + 6·11-s + (−4 − 4i)17-s + 2i·19-s + (−4 − 4i)27-s + (12 − 12i)33-s − 6·41-s + (6 − 6i)43-s + 7i·49-s − 16·51-s + (4 + 4i)57-s + 6i·59-s + (−6 − 6i)67-s + (−12 + 12i)73-s + ⋯
L(s)  = 1  + (1.15 − 1.15i)3-s − 1.66i·9-s + 1.80·11-s + (−0.970 − 0.970i)17-s + 0.458i·19-s + (−0.769 − 0.769i)27-s + (2.08 − 2.08i)33-s − 0.937·41-s + (0.914 − 0.914i)43-s + i·49-s − 2.24·51-s + (0.529 + 0.529i)57-s + 0.781i·59-s + (−0.733 − 0.733i)67-s + (−1.40 + 1.40i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86802 - 1.47838i\)
\(L(\frac12)\) \(\approx\) \(1.86802 - 1.47838i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (-2 + 2i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4 + 4i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12 - 12i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-2 + 2i)T - 83iT^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 + (12 + 12i)T + 97iT^{2} \)
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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

−9.730641213958487487680704655374, −8.993858394445167204015912707519, −8.523897225331465716576539229051, −7.37593524357020860553003023585, −6.88652404487218344817009907318, −6.01957604659786520321769656905, −4.39883158464657501388715052253, −3.39150364954002385347720469881, −2.26395011621514306972175801605, −1.20021717935956052295229915724, 1.83324053635990443842041189973, 3.17188990038395561560283720699, 4.03584833542529357139115688163, 4.63007235508842515941928406316, 6.10100292478583802497800651010, 7.03374647119733636966248631813, 8.291143655202486851025422713201, 8.887518060307621287878891217239, 9.425779081099485756661880713844, 10.25381163614466082658545874441

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