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  + 3.04·2-s + 1.42·3-s + 1.28·4-s − 10.9·5-s + 4.35·6-s − 8.62·7-s − 20.4·8-s − 24.9·9-s − 33.3·10-s + 42.2·11-s + 1.84·12-s + 78.4·13-s − 26.2·14-s − 15.6·15-s − 72.6·16-s + 75.8·17-s − 76.0·18-s + 107.·19-s − 14.1·20-s − 12.3·21-s + 128.·22-s − 27.7·23-s − 29.2·24-s − 5.15·25-s + 239.·26-s − 74.2·27-s − 11.1·28-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.275·3-s + 0.161·4-s − 0.979·5-s + 0.296·6-s − 0.465·7-s − 0.903·8-s − 0.924·9-s − 1.05·10-s + 1.15·11-s + 0.0443·12-s + 1.67·13-s − 0.501·14-s − 0.269·15-s − 1.13·16-s + 1.08·17-s − 0.995·18-s + 1.30·19-s − 0.157·20-s − 0.128·21-s + 1.24·22-s − 0.251·23-s − 0.248·24-s − 0.0412·25-s + 1.80·26-s − 0.529·27-s − 0.0749·28-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 \( 1 - 3.04T + 8T^{2} \)
3 \( 1 - 1.42T + 27T^{2} \)
5 \( 1 + 10.9T + 125T^{2} \)
7 \( 1 + 8.62T + 343T^{2} \)
11 \( 1 - 42.2T + 1.33e3T^{2} \)
13 \( 1 - 78.4T + 2.19e3T^{2} \)
17 \( 1 - 75.8T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 27.7T + 1.21e4T^{2} \)
29 \( 1 - 12.6T + 2.43e4T^{2} \)
31 \( 1 + 188.T + 2.97e4T^{2} \)
37 \( 1 - 84.0T + 5.06e4T^{2} \)
41 \( 1 + 215.T + 6.89e4T^{2} \)
47 \( 1 + 598.T + 1.03e5T^{2} \)
53 \( 1 - 368.T + 1.48e5T^{2} \)
59 \( 1 + 62.2T + 2.05e5T^{2} \)
61 \( 1 - 186.T + 2.26e5T^{2} \)
67 \( 1 + 717.T + 3.00e5T^{2} \)
71 \( 1 + 474.T + 3.57e5T^{2} \)
73 \( 1 - 1.22e3T + 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 632.T + 5.71e5T^{2} \)
89 \( 1 + 6.66T + 7.04e5T^{2} \)
97 \( 1 + 1.02e3T + 9.12e5T^{2} \)
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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

−8.506900851693786457870666797947, −7.75306398969585870074586216126, −6.64922942284120993287230044702, −5.93916948060383000142166356625, −5.26915309853831867053585440247, −4.02281871540690559756304518125, −3.52389334448573070306649435753, −3.10973081149963582967834222517, −1.30545585797618303646114879217, 0, 1.30545585797618303646114879217, 3.10973081149963582967834222517, 3.52389334448573070306649435753, 4.02281871540690559756304518125, 5.26915309853831867053585440247, 5.93916948060383000142166356625, 6.64922942284120993287230044702, 7.75306398969585870074586216126, 8.506900851693786457870666797947

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