↑

Properties

Label 2-1805-5.4-c1-0-117
Degree $2$
Conductor $1805$
Sign $0.995 + 0.0983i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  − 0.805i·2-s + 1.95i·3-s + 1.35·4-s + (2.22 + 0.219i)5-s + 1.57·6-s + 1.03i·7-s − 2.69i·8-s − 0.835·9-s + (0.177 − 1.79i)10-s + 1.80·11-s + 2.64i·12-s − 2.36i·13-s + 0.835·14-s + (−0.430 + 4.35i)15-s + 0.529·16-s − 6.49i·17-s + ⋯
L(s)  = 1  − 0.569i·2-s + 1.13i·3-s + 0.675·4-s + (0.995 + 0.0983i)5-s + 0.643·6-s + 0.391i·7-s − 0.954i·8-s − 0.278·9-s + (0.0559 − 0.566i)10-s + 0.544·11-s + 0.764i·12-s − 0.655i·13-s + 0.223·14-s + (−0.111 + 1.12i)15-s + 0.132·16-s − 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.995 + 0.0983i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.995 + 0.0983i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.802903912\)
\(L(\frac12)\) \(\approx\) \(2.802903912\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 - 0.219i)T \)
19 \( 1 \)
good2 \( 1 + 0.805iT - 2T^{2} \)
3 \( 1 - 1.95iT - 3T^{2} \)
7 \( 1 - 1.03iT - 7T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 + 6.49iT - 17T^{2} \)
23 \( 1 + 7.26iT - 23T^{2} \)
29 \( 1 - 2.22T + 29T^{2} \)
31 \( 1 - 5.25T + 31T^{2} \)
37 \( 1 + 6.67iT - 37T^{2} \)
41 \( 1 + 4.43T + 41T^{2} \)
43 \( 1 - 6.26iT - 43T^{2} \)
47 \( 1 - 8.38iT - 47T^{2} \)
53 \( 1 - 0.0601iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 6.49iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 8.23iT - 73T^{2} \)
79 \( 1 + 6.11T + 79T^{2} \)
83 \( 1 - 4.95iT - 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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

−9.357497870842113398157659427745, −9.007430868997070678541706024573, −7.66143396884125084377933711423, −6.68557567905214392975317430105, −6.03489929521572520466775610210, −5.04912182466282345112786503347, −4.29021000622685701770811451307, −3.00483879003296421905805191256, −2.57909555523292800693741940254, −1.16269307878045204387530825632, 1.56475703690445399034366841516, 1.73753476957424904375395394572, 3.13319795368005379691303534342, 4.49446430817183256110922312629, 5.71964936351343921146884202880, 6.28938073409879160233194013216, 6.79682338346475275491603440921, 7.50215954474573954427690044071, 8.283082798564629561455050319245, 9.069857245205831114466625453952

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