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  + 2.44·2-s + 8.05·3-s − 2.02·4-s + 13.2·5-s + 19.6·6-s − 1.50·7-s − 24.5·8-s + 37.8·9-s + 32.5·10-s − 19.9·11-s − 16.2·12-s − 49.4·13-s − 3.68·14-s + 107.·15-s − 43.7·16-s − 96.4·17-s + 92.6·18-s − 105.·19-s − 26.8·20-s − 12.1·21-s − 48.8·22-s − 171.·23-s − 197.·24-s + 51.7·25-s − 120.·26-s + 87.7·27-s + 3.04·28-s + ⋯
L(s)  = 1  + 0.864·2-s + 1.55·3-s − 0.252·4-s + 1.18·5-s + 1.34·6-s − 0.0813·7-s − 1.08·8-s + 1.40·9-s + 1.02·10-s − 0.547·11-s − 0.392·12-s − 1.05·13-s − 0.0703·14-s + 1.84·15-s − 0.683·16-s − 1.37·17-s + 1.21·18-s − 1.26·19-s − 0.300·20-s − 0.126·21-s − 0.473·22-s − 1.55·23-s − 1.67·24-s + 0.414·25-s − 0.911·26-s + 0.625·27-s + 0.0205·28-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 \( 1 - 2.44T + 8T^{2} \)
3 \( 1 - 8.05T + 27T^{2} \)
5 \( 1 - 13.2T + 125T^{2} \)
7 \( 1 + 1.50T + 343T^{2} \)
11 \( 1 + 19.9T + 1.33e3T^{2} \)
13 \( 1 + 49.4T + 2.19e3T^{2} \)
17 \( 1 + 96.4T + 4.91e3T^{2} \)
19 \( 1 + 105.T + 6.85e3T^{2} \)
23 \( 1 + 171.T + 1.21e4T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 - 91.9T + 2.97e4T^{2} \)
37 \( 1 + 104.T + 5.06e4T^{2} \)
41 \( 1 + 407.T + 6.89e4T^{2} \)
47 \( 1 - 482.T + 1.03e5T^{2} \)
53 \( 1 + 698.T + 1.48e5T^{2} \)
59 \( 1 - 338.T + 2.05e5T^{2} \)
61 \( 1 - 330.T + 2.26e5T^{2} \)
67 \( 1 - 307.T + 3.00e5T^{2} \)
71 \( 1 - 49.0T + 3.57e5T^{2} \)
73 \( 1 + 919.T + 3.89e5T^{2} \)
79 \( 1 - 682.T + 4.93e5T^{2} \)
83 \( 1 - 411.T + 5.71e5T^{2} \)
89 \( 1 - 764.T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−8.602821235969831935158919956490, −7.941103922072089126759804662351, −6.69625549466020915839422191558, −6.10567744915705066656478340834, −4.94967119443380911249995379866, −4.38072816741456706895742606256, −3.36309615618246091281846835706, −2.33861966946649757010822296785, −2.12426428011668221928143594081, 0, 2.12426428011668221928143594081, 2.33861966946649757010822296785, 3.36309615618246091281846835706, 4.38072816741456706895742606256, 4.94967119443380911249995379866, 6.10567744915705066656478340834, 6.69625549466020915839422191558, 7.941103922072089126759804662351, 8.602821235969831935158919956490

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