Properties

Label 2-43e2-1.1-c3-0-245
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.25·2-s − 4.02·3-s − 2.93·4-s − 4.98·5-s − 9.06·6-s − 5.25·7-s − 24.6·8-s − 10.7·9-s − 11.2·10-s + 25.9·11-s + 11.8·12-s + 14.2·13-s − 11.8·14-s + 20.0·15-s − 31.9·16-s + 20.3·17-s − 24.2·18-s + 2.21·19-s + 14.6·20-s + 21.1·21-s + 58.4·22-s + 39.1·23-s + 99.1·24-s − 100.·25-s + 31.9·26-s + 152.·27-s + 15.3·28-s + ⋯
L(s)  = 1  + 0.795·2-s − 0.775·3-s − 0.366·4-s − 0.446·5-s − 0.616·6-s − 0.283·7-s − 1.08·8-s − 0.399·9-s − 0.355·10-s + 0.711·11-s + 0.284·12-s + 0.303·13-s − 0.225·14-s + 0.345·15-s − 0.499·16-s + 0.290·17-s − 0.317·18-s + 0.0267·19-s + 0.163·20-s + 0.219·21-s + 0.566·22-s + 0.354·23-s + 0.843·24-s − 0.800·25-s + 0.241·26-s + 1.08·27-s + 0.103·28-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: $\chi_{1849} (1, \cdot )$
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 \( 1 - 2.25T + 8T^{2} \)
3 \( 1 + 4.02T + 27T^{2} \)
5 \( 1 + 4.98T + 125T^{2} \)
7 \( 1 + 5.25T + 343T^{2} \)
11 \( 1 - 25.9T + 1.33e3T^{2} \)
13 \( 1 - 14.2T + 2.19e3T^{2} \)
17 \( 1 - 20.3T + 4.91e3T^{2} \)
19 \( 1 - 2.21T + 6.85e3T^{2} \)
23 \( 1 - 39.1T + 1.21e4T^{2} \)
29 \( 1 - 277.T + 2.43e4T^{2} \)
31 \( 1 + 47.2T + 2.97e4T^{2} \)
37 \( 1 - 43.9T + 5.06e4T^{2} \)
41 \( 1 - 98.7T + 6.89e4T^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + 230.T + 1.48e5T^{2} \)
59 \( 1 + 400.T + 2.05e5T^{2} \)
61 \( 1 - 13.3T + 2.26e5T^{2} \)
67 \( 1 + 640.T + 3.00e5T^{2} \)
71 \( 1 - 251.T + 3.57e5T^{2} \)
73 \( 1 + 526.T + 3.89e5T^{2} \)
79 \( 1 - 978.T + 4.93e5T^{2} \)
83 \( 1 + 37.2T + 5.71e5T^{2} \)
89 \( 1 - 437.T + 7.04e5T^{2} \)
97 \( 1 + 1.12e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.591164466858738421611342889865, −7.64080146739220386019652847421, −6.44050430492123356409189843610, −6.09172966612277867247623052702, −5.17450000649227853547351914228, −4.44316952290746808682489055716, −3.62276132536920435398048010830, −2.77944483924923183773076615671, −1.01548886004882102303929666389, 0, 1.01548886004882102303929666389, 2.77944483924923183773076615671, 3.62276132536920435398048010830, 4.44316952290746808682489055716, 5.17450000649227853547351914228, 6.09172966612277867247623052702, 6.44050430492123356409189843610, 7.64080146739220386019652847421, 8.591164466858738421611342889865

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