Properties

Label 2-43e2-1.1-c3-0-184
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.22·2-s − 8.71·3-s + 2.37·4-s + 13.5·5-s + 28.0·6-s − 24.0·7-s + 18.1·8-s + 49.0·9-s − 43.5·10-s − 71.6·11-s − 20.6·12-s + 26.7·13-s + 77.5·14-s − 117.·15-s − 77.3·16-s + 35.6·17-s − 157.·18-s + 88.4·19-s + 32.0·20-s + 209.·21-s + 230.·22-s − 64.6·23-s − 158.·24-s + 57.6·25-s − 86.2·26-s − 192.·27-s − 57.1·28-s + ⋯
L(s)  = 1  − 1.13·2-s − 1.67·3-s + 0.296·4-s + 1.20·5-s + 1.91·6-s − 1.29·7-s + 0.800·8-s + 1.81·9-s − 1.37·10-s − 1.96·11-s − 0.497·12-s + 0.571·13-s + 1.47·14-s − 2.02·15-s − 1.20·16-s + 0.508·17-s − 2.06·18-s + 1.06·19-s + 0.358·20-s + 2.18·21-s + 2.23·22-s − 0.586·23-s − 1.34·24-s + 0.461·25-s − 0.650·26-s − 1.36·27-s − 0.385·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + 3.22T + 8T^{2} \)
3 \( 1 + 8.71T + 27T^{2} \)
5 \( 1 - 13.5T + 125T^{2} \)
7 \( 1 + 24.0T + 343T^{2} \)
11 \( 1 + 71.6T + 1.33e3T^{2} \)
13 \( 1 - 26.7T + 2.19e3T^{2} \)
17 \( 1 - 35.6T + 4.91e3T^{2} \)
19 \( 1 - 88.4T + 6.85e3T^{2} \)
23 \( 1 + 64.6T + 1.21e4T^{2} \)
29 \( 1 - 106.T + 2.43e4T^{2} \)
31 \( 1 + 7.27T + 2.97e4T^{2} \)
37 \( 1 + 195.T + 5.06e4T^{2} \)
41 \( 1 + 404.T + 6.89e4T^{2} \)
47 \( 1 - 31.1T + 1.03e5T^{2} \)
53 \( 1 - 34.4T + 1.48e5T^{2} \)
59 \( 1 + 486.T + 2.05e5T^{2} \)
61 \( 1 - 491.T + 2.26e5T^{2} \)
67 \( 1 - 102.T + 3.00e5T^{2} \)
71 \( 1 + 119.T + 3.57e5T^{2} \)
73 \( 1 - 123.T + 3.89e5T^{2} \)
79 \( 1 - 34.0T + 4.93e5T^{2} \)
83 \( 1 + 681.T + 5.71e5T^{2} \)
89 \( 1 - 1.56e3T + 7.04e5T^{2} \)
97 \( 1 - 93.1T + 9.12e5T^{2} \)
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

−8.635751233199801262911262442719, −7.63533921952729762012391382296, −6.85271568399155428018827886710, −6.05931645094097434436167655778, −5.46745960550115079133874997074, −4.84522169772652181644567435615, −3.24753298602404359020814448209, −1.92875544201684334978557938393, −0.792991629104931123358059368856, 0, 0.792991629104931123358059368856, 1.92875544201684334978557938393, 3.24753298602404359020814448209, 4.84522169772652181644567435615, 5.46745960550115079133874997074, 6.05931645094097434436167655778, 6.85271568399155428018827886710, 7.63533921952729762012391382296, 8.635751233199801262911262442719

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