↑

Properties

Label 2-3800-5.4-c1-0-25
Degree $2$
Conductor $3800$
Sign $-0.447 - 0.894i$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
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  + 1.83i·3-s + 1.83i·7-s − 0.364·9-s + 0.834·11-s − 2.19i·13-s − 2.56i·17-s + 19-s − 3.36·21-s + 0.635i·23-s + 4.83i·27-s + 9.62·29-s − 6.59·31-s + 1.53i·33-s + 5.23i·37-s + 4.03·39-s + ⋯
L(s)  = 1  + 1.05i·3-s + 0.693i·7-s − 0.121·9-s + 0.251·11-s − 0.609i·13-s − 0.621i·17-s + 0.229·19-s − 0.734·21-s + 0.132i·23-s + 0.930i·27-s + 1.78·29-s − 1.18·31-s + 0.266i·33-s + 0.860i·37-s + 0.645·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.861924536\)
\(L(\frac12)\) \(\approx\) \(1.861924536\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 1.83iT - 3T^{2} \)
7 \( 1 - 1.83iT - 7T^{2} \)
11 \( 1 - 0.834T + 11T^{2} \)
13 \( 1 + 2.19iT - 13T^{2} \)
17 \( 1 + 2.56iT - 17T^{2} \)
23 \( 1 - 0.635iT - 23T^{2} \)
29 \( 1 - 9.62T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 - 5.23iT - 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 - 7.06iT - 43T^{2} \)
47 \( 1 - 9.86iT - 47T^{2} \)
53 \( 1 + 0.668iT - 53T^{2} \)
59 \( 1 + 0.397T + 59T^{2} \)
61 \( 1 - 2.26T + 61T^{2} \)
67 \( 1 - 2.43iT - 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 9.49iT - 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 - 5.97T + 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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

−8.908489495403596880836706590520, −8.131338473279670910477379655470, −7.31569831751891874675111182029, −6.41100390149538798418151000899, −5.59010999425794326914001971437, −4.91178129373782704811902515446, −4.25921993980672892949589834630, −3.27285231820678467931274711395, −2.60205291545495832822974439348, −1.16512632568380846914502097557, 0.61399921636274559247474804549, 1.58310955005510152360629385083, 2.40862489649484407571771670361, 3.69211282472426697328912494887, 4.30421526036297072065023350168, 5.36970284990587462464783735011, 6.29000741570215409440299144727, 6.90597384204016418864381360902, 7.35531945665832192930781275913, 8.153247589893984042804014398983

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