Properties

Label 2-3200-40.29-c1-0-47
Degree $2$
Conductor $3200$
Sign $0.894 - 0.447i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.14·3-s + 6.89·9-s + 6.61i·11-s − 7.89i·17-s + 2.51i·19-s + 12.2·27-s + 20.7i·33-s + 12.7·41-s + 8.48·43-s + 7·49-s − 24.8i·51-s + 7.89i·57-s + 14.1i·59-s − 7.88·67-s + 13.6i·73-s + ⋯
L(s)  = 1  + 1.81·3-s + 2.29·9-s + 1.99i·11-s − 1.91i·17-s + 0.575i·19-s + 2.36·27-s + 3.62i·33-s + 1.99·41-s + 1.29·43-s + 49-s − 3.48i·51-s + 1.04i·57-s + 1.84i·59-s − 0.962·67-s + 1.60i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.872015880\)
\(L(\frac12)\) \(\approx\) \(3.872015880\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.14T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 6.61iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.89iT - 17T^{2} \)
19 \( 1 - 2.51iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 12.7T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 7.88T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.870942499135906625028042674216, −7.87484234375644257575952676319, −7.26655784388676931346745391755, −7.07835997738194773346562501417, −5.58681808862167487275126151904, −4.39991465280719747876510438221, −4.16819083496210740853678949335, −2.77953757995571206974801453813, −2.46096172149270170728816452702, −1.35739361210159566765396160787, 1.04915517856324537507270032164, 2.20462981083968075415354924776, 3.03850540824081876243881689517, 3.70056966729130069987874686345, 4.34128428266210633565032872472, 5.74043306232134414332336472567, 6.35748788928443585368827483395, 7.47509736126888225559663828783, 8.030979918585618768487476142105, 8.694456966113265450858356815617

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