Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34·2-s + 2.42·3-s − 2.50·4-s − 18.7·5-s − 5.69·6-s − 0.512·7-s + 24.6·8-s − 21.1·9-s + 43.8·10-s + 5.04·11-s − 6.07·12-s − 48.6·13-s + 1.20·14-s − 45.4·15-s − 37.7·16-s − 6.35·17-s + 49.4·18-s − 123.·19-s + 46.7·20-s − 1.24·21-s − 11.8·22-s + 141.·23-s + 59.8·24-s + 224.·25-s + 114.·26-s − 116.·27-s + 1.28·28-s + ⋯
L(s)  = 1  − 0.829·2-s + 0.467·3-s − 0.312·4-s − 1.67·5-s − 0.387·6-s − 0.0276·7-s + 1.08·8-s − 0.781·9-s + 1.38·10-s + 0.138·11-s − 0.146·12-s − 1.03·13-s + 0.0229·14-s − 0.782·15-s − 0.589·16-s − 0.0907·17-s + 0.647·18-s − 1.49·19-s + 0.523·20-s − 0.0129·21-s − 0.114·22-s + 1.27·23-s + 0.508·24-s + 1.79·25-s + 0.860·26-s − 0.832·27-s + 0.00865·28-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 \( 1 + 2.34T + 8T^{2} \)
3 \( 1 - 2.42T + 27T^{2} \)
5 \( 1 + 18.7T + 125T^{2} \)
7 \( 1 + 0.512T + 343T^{2} \)
11 \( 1 - 5.04T + 1.33e3T^{2} \)
13 \( 1 + 48.6T + 2.19e3T^{2} \)
17 \( 1 + 6.35T + 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 - 141.T + 1.21e4T^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 - 164.T + 2.97e4T^{2} \)
37 \( 1 - 365.T + 5.06e4T^{2} \)
41 \( 1 + 175.T + 6.89e4T^{2} \)
47 \( 1 - 219.T + 1.03e5T^{2} \)
53 \( 1 - 34.4T + 1.48e5T^{2} \)
59 \( 1 + 771.T + 2.05e5T^{2} \)
61 \( 1 - 432.T + 2.26e5T^{2} \)
67 \( 1 + 644.T + 3.00e5T^{2} \)
71 \( 1 - 919.T + 3.57e5T^{2} \)
73 \( 1 + 98.7T + 3.89e5T^{2} \)
79 \( 1 - 411.T + 4.93e5T^{2} \)
83 \( 1 - 271.T + 5.71e5T^{2} \)
89 \( 1 + 115.T + 7.04e5T^{2} \)
97 \( 1 - 623.T + 9.12e5T^{2} \)
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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

−8.448941602571792483466309999698, −7.980951388856467486356250764998, −7.30707194205630300731220285329, −6.40312291997531260585002257755, −4.80939668453872449982068670141, −4.45359340703033336659572491397, −3.37115338445034228305952352725, −2.45433127301252791915340500962, −0.814517697467941665348562855808, 0, 0.814517697467941665348562855808, 2.45433127301252791915340500962, 3.37115338445034228305952352725, 4.45359340703033336659572491397, 4.80939668453872449982068670141, 6.40312291997531260585002257755, 7.30707194205630300731220285329, 7.980951388856467486356250764998, 8.448941602571792483466309999698

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