Properties

Label 2-1045-1.1-c3-0-49
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.34·2-s − 2.02·3-s + 3.17·4-s + 5·5-s − 6.76·6-s − 13.1·7-s − 16.1·8-s − 22.9·9-s + 16.7·10-s − 11·11-s − 6.42·12-s + 0.857·13-s − 44.1·14-s − 10.1·15-s − 79.3·16-s + 110.·17-s − 76.5·18-s + 19·19-s + 15.8·20-s + 26.7·21-s − 36.7·22-s − 116.·23-s + 32.6·24-s + 25·25-s + 2.86·26-s + 101.·27-s − 41.8·28-s + ⋯
L(s)  = 1  + 1.18·2-s − 0.389·3-s + 0.396·4-s + 0.447·5-s − 0.460·6-s − 0.712·7-s − 0.712·8-s − 0.848·9-s + 0.528·10-s − 0.301·11-s − 0.154·12-s + 0.0183·13-s − 0.841·14-s − 0.174·15-s − 1.23·16-s + 1.57·17-s − 1.00·18-s + 0.229·19-s + 0.177·20-s + 0.277·21-s − 0.356·22-s − 1.05·23-s + 0.277·24-s + 0.200·25-s + 0.0216·26-s + 0.720·27-s − 0.282·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.530893510\)
\(L(\frac12)\) \(\approx\) \(2.530893510\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5T \)
11 \( 1 + 11T \)
19 \( 1 - 19T \)
good2 \( 1 - 3.34T + 8T^{2} \)
3 \( 1 + 2.02T + 27T^{2} \)
7 \( 1 + 13.1T + 343T^{2} \)
13 \( 1 - 0.857T + 2.19e3T^{2} \)
17 \( 1 - 110.T + 4.91e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 - 208.T + 2.97e4T^{2} \)
37 \( 1 - 239.T + 5.06e4T^{2} \)
41 \( 1 + 50.2T + 6.89e4T^{2} \)
43 \( 1 - 190.T + 7.95e4T^{2} \)
47 \( 1 - 417.T + 1.03e5T^{2} \)
53 \( 1 - 76.7T + 1.48e5T^{2} \)
59 \( 1 - 128.T + 2.05e5T^{2} \)
61 \( 1 - 123.T + 2.26e5T^{2} \)
67 \( 1 + 650.T + 3.00e5T^{2} \)
71 \( 1 - 883.T + 3.57e5T^{2} \)
73 \( 1 + 872.T + 3.89e5T^{2} \)
79 \( 1 - 815.T + 4.93e5T^{2} \)
83 \( 1 + 504.T + 5.71e5T^{2} \)
89 \( 1 + 745.T + 7.04e5T^{2} \)
97 \( 1 + 906.T + 9.12e5T^{2} \)
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

−9.732591949985092992490124160012, −8.738399380610000130952000468716, −7.79437216157980222530151325185, −6.49189288476640689133766628366, −5.92122213850468058999651426998, −5.36625312969225862782251174183, −4.35302806505392381789348780247, −3.24373026286192802614758181989, −2.59384058537226824849952431893, −0.69915031355808590104780307835, 0.69915031355808590104780307835, 2.59384058537226824849952431893, 3.24373026286192802614758181989, 4.35302806505392381789348780247, 5.36625312969225862782251174183, 5.92122213850468058999651426998, 6.49189288476640689133766628366, 7.79437216157980222530151325185, 8.738399380610000130952000468716, 9.732591949985092992490124160012

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