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  + 3.90·2-s − 3.52·3-s + 7.24·4-s + 10.7·5-s − 13.7·6-s − 28.1·7-s − 2.93·8-s − 14.6·9-s + 41.8·10-s + 63.0·11-s − 25.5·12-s − 23.3·13-s − 109.·14-s − 37.6·15-s − 69.4·16-s + 49.0·17-s − 57.0·18-s + 57.3·19-s + 77.6·20-s + 99.0·21-s + 246.·22-s + 137.·23-s + 10.3·24-s − 10.3·25-s − 91.2·26-s + 146.·27-s − 204.·28-s + ⋯
L(s)  = 1  + 1.38·2-s − 0.677·3-s + 0.906·4-s + 0.957·5-s − 0.935·6-s − 1.51·7-s − 0.129·8-s − 0.541·9-s + 1.32·10-s + 1.72·11-s − 0.613·12-s − 0.498·13-s − 2.09·14-s − 0.648·15-s − 1.08·16-s + 0.699·17-s − 0.746·18-s + 0.691·19-s + 0.867·20-s + 1.02·21-s + 2.38·22-s + 1.25·23-s + 0.0878·24-s − 0.0831·25-s − 0.687·26-s + 1.04·27-s − 1.37·28-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 \( 1 - 3.90T + 8T^{2} \)
3 \( 1 + 3.52T + 27T^{2} \)
5 \( 1 - 10.7T + 125T^{2} \)
7 \( 1 + 28.1T + 343T^{2} \)
11 \( 1 - 63.0T + 1.33e3T^{2} \)
13 \( 1 + 23.3T + 2.19e3T^{2} \)
17 \( 1 - 49.0T + 4.91e3T^{2} \)
19 \( 1 - 57.3T + 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 + 69.2T + 2.97e4T^{2} \)
37 \( 1 + 51.1T + 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
47 \( 1 + 432.T + 1.03e5T^{2} \)
53 \( 1 + 753.T + 1.48e5T^{2} \)
59 \( 1 + 845.T + 2.05e5T^{2} \)
61 \( 1 + 156.T + 2.26e5T^{2} \)
67 \( 1 - 618.T + 3.00e5T^{2} \)
71 \( 1 + 723.T + 3.57e5T^{2} \)
73 \( 1 + 315.T + 3.89e5T^{2} \)
79 \( 1 + 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 587.T + 5.71e5T^{2} \)
89 \( 1 + 1.07e3T + 7.04e5T^{2} \)
97 \( 1 + 402.T + 9.12e5T^{2} \)
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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

−8.761350832810036292491367271433, −7.07183209917727601643687791983, −6.45440098791280627570856950246, −6.04912767967583477042588271345, −5.35262154970641477246839089445, −4.51782060882787475846676728802, −3.27682774947987873029682188594, −3.00361983863989753590003388225, −1.41918676701163934048213622079, 0, 1.41918676701163934048213622079, 3.00361983863989753590003388225, 3.27682774947987873029682188594, 4.51782060882787475846676728802, 5.35262154970641477246839089445, 6.04912767967583477042588271345, 6.45440098791280627570856950246, 7.07183209917727601643687791983, 8.761350832810036292491367271433

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