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  + 3.81·2-s − 8.34·3-s + 6.58·4-s − 2.56·5-s − 31.8·6-s + 3.42·7-s − 5.39·8-s + 42.6·9-s − 9.79·10-s − 24.5·11-s − 54.9·12-s − 46.2·13-s + 13.0·14-s + 21.4·15-s − 73.3·16-s + 97.5·17-s + 163.·18-s + 131.·19-s − 16.8·20-s − 28.5·21-s − 93.6·22-s + 175.·23-s + 45.0·24-s − 118.·25-s − 176.·26-s − 130.·27-s + 22.5·28-s + ⋯
L(s)  = 1  + 1.35·2-s − 1.60·3-s + 0.823·4-s − 0.229·5-s − 2.16·6-s + 0.184·7-s − 0.238·8-s + 1.58·9-s − 0.309·10-s − 0.672·11-s − 1.32·12-s − 0.987·13-s + 0.249·14-s + 0.368·15-s − 1.14·16-s + 1.39·17-s + 2.13·18-s + 1.59·19-s − 0.188·20-s − 0.296·21-s − 0.907·22-s + 1.58·23-s + 0.383·24-s − 0.947·25-s − 1.33·26-s − 0.933·27-s + 0.152·28-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 \( 1 - 3.81T + 8T^{2} \)
3 \( 1 + 8.34T + 27T^{2} \)
5 \( 1 + 2.56T + 125T^{2} \)
7 \( 1 - 3.42T + 343T^{2} \)
11 \( 1 + 24.5T + 1.33e3T^{2} \)
13 \( 1 + 46.2T + 2.19e3T^{2} \)
17 \( 1 - 97.5T + 4.91e3T^{2} \)
19 \( 1 - 131.T + 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 - 83.1T + 2.43e4T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 + 301.T + 5.06e4T^{2} \)
41 \( 1 + 201.T + 6.89e4T^{2} \)
47 \( 1 + 335.T + 1.03e5T^{2} \)
53 \( 1 - 373.T + 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 - 892.T + 2.26e5T^{2} \)
67 \( 1 - 2.66T + 3.00e5T^{2} \)
71 \( 1 + 467.T + 3.57e5T^{2} \)
73 \( 1 - 481.T + 3.89e5T^{2} \)
79 \( 1 + 296.T + 4.93e5T^{2} \)
83 \( 1 - 122.T + 5.71e5T^{2} \)
89 \( 1 + 746.T + 7.04e5T^{2} \)
97 \( 1 + 1.72e3T + 9.12e5T^{2} \)
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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

−8.253159717801776557482735038664, −7.21712038513936969052316254164, −6.73970681291910471233921341133, −5.53981246647270682894707488601, −5.28235654565260329157856797537, −4.81714865040389882435947486198, −3.64367293317627771040631542755, −2.77065503434456314285324853132, −1.13634254768471285528160684803, 0, 1.13634254768471285528160684803, 2.77065503434456314285324853132, 3.64367293317627771040631542755, 4.81714865040389882435947486198, 5.28235654565260329157856797537, 5.53981246647270682894707488601, 6.73970681291910471233921341133, 7.21712038513936969052316254164, 8.253159717801776557482735038664

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