Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 5.46·2-s + 9.09·3-s + 21.8·4-s − 6.15·5-s − 49.6·6-s + 2.59·7-s − 75.5·8-s + 55.7·9-s + 33.6·10-s − 20.0·11-s + 198.·12-s − 34.1·13-s − 14.1·14-s − 55.9·15-s + 238.·16-s + 36.1·17-s − 304.·18-s + 142.·19-s − 134.·20-s + 23.5·21-s + 109.·22-s − 132.·23-s − 687.·24-s − 87.1·25-s + 186.·26-s + 261.·27-s + 56.5·28-s + ⋯
L(s)  = 1  − 1.93·2-s + 1.75·3-s + 2.72·4-s − 0.550·5-s − 3.38·6-s + 0.139·7-s − 3.33·8-s + 2.06·9-s + 1.06·10-s − 0.549·11-s + 4.77·12-s − 0.728·13-s − 0.270·14-s − 0.963·15-s + 3.72·16-s + 0.515·17-s − 3.98·18-s + 1.72·19-s − 1.50·20-s + 0.244·21-s + 1.06·22-s − 1.20·23-s − 5.84·24-s − 0.697·25-s + 1.40·26-s + 1.86·27-s + 0.381·28-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: $-1$
Motivic weight: \(3\)
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 \( 1 + 5.46T + 8T^{2} \)
3 \( 1 - 9.09T + 27T^{2} \)
5 \( 1 + 6.15T + 125T^{2} \)
7 \( 1 - 2.59T + 343T^{2} \)
11 \( 1 + 20.0T + 1.33e3T^{2} \)
13 \( 1 + 34.1T + 2.19e3T^{2} \)
17 \( 1 - 36.1T + 4.91e3T^{2} \)
19 \( 1 - 142.T + 6.85e3T^{2} \)
23 \( 1 + 132.T + 1.21e4T^{2} \)
29 \( 1 + 49.6T + 2.43e4T^{2} \)
31 \( 1 + 51.3T + 2.97e4T^{2} \)
37 \( 1 + 49.1T + 5.06e4T^{2} \)
41 \( 1 - 71.4T + 6.89e4T^{2} \)
47 \( 1 + 371.T + 1.03e5T^{2} \)
53 \( 1 - 260.T + 1.48e5T^{2} \)
59 \( 1 - 29.4T + 2.05e5T^{2} \)
61 \( 1 + 461.T + 2.26e5T^{2} \)
67 \( 1 - 891.T + 3.00e5T^{2} \)
71 \( 1 - 475.T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + 307.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + 670.T + 7.04e5T^{2} \)
97 \( 1 + 125.T + 9.12e5T^{2} \)
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

−8.287037622505824148976406923560, −7.947028555890248893663136236996, −7.55229794238710957143582668439, −6.80866558306230609641796970357, −5.41754265509872459396425559111, −3.77688838906044674325461372638, −2.99553499090269208073357351681, −2.21281644232752404284207230695, −1.33934434828812913523043084276, 0, 1.33934434828812913523043084276, 2.21281644232752404284207230695, 2.99553499090269208073357351681, 3.77688838906044674325461372638, 5.41754265509872459396425559111, 6.80866558306230609641796970357, 7.55229794238710957143582668439, 7.947028555890248893663136236996, 8.287037622505824148976406923560

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