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  − 0.618·2-s − 9.18·3-s − 7.61·4-s + 19.4·5-s + 5.67·6-s + 15.1·7-s + 9.65·8-s + 57.2·9-s − 12.0·10-s − 7.19·11-s + 69.9·12-s + 36.4·13-s − 9.37·14-s − 178.·15-s + 54.9·16-s + 102.·17-s − 35.4·18-s − 51.6·19-s − 147.·20-s − 139.·21-s + 4.44·22-s − 102.·23-s − 88.6·24-s + 252.·25-s − 22.5·26-s − 277.·27-s − 115.·28-s + ⋯
L(s)  = 1  − 0.218·2-s − 1.76·3-s − 0.952·4-s + 1.73·5-s + 0.386·6-s + 0.818·7-s + 0.426·8-s + 2.12·9-s − 0.379·10-s − 0.197·11-s + 1.68·12-s + 0.777·13-s − 0.178·14-s − 3.06·15-s + 0.858·16-s + 1.45·17-s − 0.463·18-s − 0.624·19-s − 1.65·20-s − 1.44·21-s + 0.0431·22-s − 0.929·23-s − 0.753·24-s + 2.01·25-s − 0.170·26-s − 1.98·27-s − 0.779·28-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 \( 1 + 0.618T + 8T^{2} \)
3 \( 1 + 9.18T + 27T^{2} \)
5 \( 1 - 19.4T + 125T^{2} \)
7 \( 1 - 15.1T + 343T^{2} \)
11 \( 1 + 7.19T + 1.33e3T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 + 51.6T + 6.85e3T^{2} \)
23 \( 1 + 102.T + 1.21e4T^{2} \)
29 \( 1 + 96.3T + 2.43e4T^{2} \)
31 \( 1 + 157.T + 2.97e4T^{2} \)
37 \( 1 + 233.T + 5.06e4T^{2} \)
41 \( 1 + 39.6T + 6.89e4T^{2} \)
47 \( 1 + 455.T + 1.03e5T^{2} \)
53 \( 1 + 602.T + 1.48e5T^{2} \)
59 \( 1 + 124.T + 2.05e5T^{2} \)
61 \( 1 - 838.T + 2.26e5T^{2} \)
67 \( 1 + 166.T + 3.00e5T^{2} \)
71 \( 1 - 215.T + 3.57e5T^{2} \)
73 \( 1 + 108.T + 3.89e5T^{2} \)
79 \( 1 + 494.T + 4.93e5T^{2} \)
83 \( 1 - 712.T + 5.71e5T^{2} \)
89 \( 1 + 431.T + 7.04e5T^{2} \)
97 \( 1 - 1.29e3T + 9.12e5T^{2} \)
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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

−8.589129943717091093411669528754, −7.71040766087249027722448633037, −6.54338214348394639875568169935, −5.86342057853144007979042290810, −5.32766904040861112922327687501, −4.87455836036078955995958336530, −3.70667130744666053619056843778, −1.73940532584875350653739750030, −1.26550482555400419440134412325, 0, 1.26550482555400419440134412325, 1.73940532584875350653739750030, 3.70667130744666053619056843778, 4.87455836036078955995958336530, 5.32766904040861112922327687501, 5.86342057853144007979042290810, 6.54338214348394639875568169935, 7.71040766087249027722448633037, 8.589129943717091093411669528754

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