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  − 5.40·2-s − 2.51·3-s + 21.2·4-s − 3.24·5-s + 13.5·6-s + 35.6·7-s − 71.4·8-s − 20.6·9-s + 17.5·10-s − 42.5·11-s − 53.3·12-s + 22.6·13-s − 192.·14-s + 8.17·15-s + 216.·16-s − 6.66·17-s + 111.·18-s + 14.7·19-s − 68.9·20-s − 89.5·21-s + 230.·22-s − 95.0·23-s + 179.·24-s − 114.·25-s − 122.·26-s + 119.·27-s + 755.·28-s + ⋯
L(s)  = 1  − 1.91·2-s − 0.484·3-s + 2.65·4-s − 0.290·5-s + 0.925·6-s + 1.92·7-s − 3.15·8-s − 0.765·9-s + 0.555·10-s − 1.16·11-s − 1.28·12-s + 0.482·13-s − 3.67·14-s + 0.140·15-s + 3.38·16-s − 0.0950·17-s + 1.46·18-s + 0.178·19-s − 0.770·20-s − 0.930·21-s + 2.22·22-s − 0.861·23-s + 1.52·24-s − 0.915·25-s − 0.922·26-s + 0.854·27-s + 5.10·28-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 \( 1 + 5.40T + 8T^{2} \)
3 \( 1 + 2.51T + 27T^{2} \)
5 \( 1 + 3.24T + 125T^{2} \)
7 \( 1 - 35.6T + 343T^{2} \)
11 \( 1 + 42.5T + 1.33e3T^{2} \)
13 \( 1 - 22.6T + 2.19e3T^{2} \)
17 \( 1 + 6.66T + 4.91e3T^{2} \)
19 \( 1 - 14.7T + 6.85e3T^{2} \)
23 \( 1 + 95.0T + 1.21e4T^{2} \)
29 \( 1 - 125.T + 2.43e4T^{2} \)
31 \( 1 - 56.0T + 2.97e4T^{2} \)
37 \( 1 - 217.T + 5.06e4T^{2} \)
41 \( 1 - 342.T + 6.89e4T^{2} \)
47 \( 1 + 399.T + 1.03e5T^{2} \)
53 \( 1 - 211.T + 1.48e5T^{2} \)
59 \( 1 + 435.T + 2.05e5T^{2} \)
61 \( 1 + 693.T + 2.26e5T^{2} \)
67 \( 1 + 141.T + 3.00e5T^{2} \)
71 \( 1 - 632.T + 3.57e5T^{2} \)
73 \( 1 + 736.T + 3.89e5T^{2} \)
79 \( 1 - 165.T + 4.93e5T^{2} \)
83 \( 1 + 1.20e3T + 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 - 625.T + 9.12e5T^{2} \)
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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

−8.262086154913809568190757200632, −8.011438874649167840342455881400, −7.45694950399947060873493712914, −6.20453240755180733756967602404, −5.57980576853202464972465728492, −4.51142014674620539761509749797, −2.84538145871179571244995534854, −1.98639293925761198857277056721, −1.01245461134464940880511470420, 0, 1.01245461134464940880511470420, 1.98639293925761198857277056721, 2.84538145871179571244995534854, 4.51142014674620539761509749797, 5.57980576853202464972465728492, 6.20453240755180733756967602404, 7.45694950399947060873493712914, 8.011438874649167840342455881400, 8.262086154913809568190757200632

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