Properties

Label 2-1805-1.1-c1-0-9
Degree $2$
Conductor $1805$
Sign $1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.381·2-s + 0.381·3-s − 1.85·4-s − 5-s + 0.145·6-s − 4.23·7-s − 1.47·8-s − 2.85·9-s − 0.381·10-s − 2.38·11-s − 0.708·12-s + 5·13-s − 1.61·14-s − 0.381·15-s + 3.14·16-s − 6·17-s − 1.09·18-s + 1.85·20-s − 1.61·21-s − 0.909·22-s + 0.618·23-s − 0.562·24-s + 25-s + 1.90·26-s − 2.23·27-s + 7.85·28-s + 4.85·29-s + ⋯
L(s)  = 1  + 0.270·2-s + 0.220·3-s − 0.927·4-s − 0.447·5-s + 0.0595·6-s − 1.60·7-s − 0.520·8-s − 0.951·9-s − 0.120·10-s − 0.718·11-s − 0.204·12-s + 1.38·13-s − 0.432·14-s − 0.0986·15-s + 0.786·16-s − 1.45·17-s − 0.256·18-s + 0.414·20-s − 0.353·21-s − 0.193·22-s + 0.128·23-s − 0.114·24-s + 0.200·25-s + 0.374·26-s − 0.430·27-s + 1.48·28-s + 0.901·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.7325148270\)
\(L(\frac12)\) \(\approx\) \(0.7325148270\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - 0.381T + 2T^{2} \)
3 \( 1 - 0.381T + 3T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 + 2.38T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
23 \( 1 - 0.618T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 - 4.85T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 + 5.76T + 47T^{2} \)
53 \( 1 - 3.38T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4.47T + 83T^{2} \)
89 \( 1 - 6.70T + 89T^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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

−9.122541929476331572624286948149, −8.546786914046459803681617297868, −8.017593115350204332822419870162, −6.49231130161351995652205324752, −6.27352221603009970136942859038, −5.12835402075422783857760334767, −4.19476789550799328494919023721, −3.32766844767614596954852758652, −2.74243261929158210873429639762, −0.52979928575737552900917647218, 0.52979928575737552900917647218, 2.74243261929158210873429639762, 3.32766844767614596954852758652, 4.19476789550799328494919023721, 5.12835402075422783857760334767, 6.27352221603009970136942859038, 6.49231130161351995652205324752, 8.017593115350204332822419870162, 8.546786914046459803681617297868, 9.122541929476331572624286948149

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