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  − 0.928·2-s + 4.15·3-s − 7.13·4-s − 7.22·5-s − 3.86·6-s − 33.4·7-s + 14.0·8-s − 9.70·9-s + 6.70·10-s + 21.8·11-s − 29.6·12-s − 37.7·13-s + 31.1·14-s − 30.0·15-s + 44.0·16-s + 87.4·17-s + 9.01·18-s + 115.·19-s + 51.5·20-s − 139.·21-s − 20.2·22-s − 3.50·23-s + 58.4·24-s − 72.8·25-s + 35.1·26-s − 152.·27-s + 239.·28-s + ⋯
L(s)  = 1  − 0.328·2-s + 0.800·3-s − 0.892·4-s − 0.645·5-s − 0.262·6-s − 1.80·7-s + 0.621·8-s − 0.359·9-s + 0.212·10-s + 0.597·11-s − 0.714·12-s − 0.806·13-s + 0.593·14-s − 0.516·15-s + 0.688·16-s + 1.24·17-s + 0.118·18-s + 1.39·19-s + 0.576·20-s − 1.44·21-s − 0.196·22-s − 0.0317·23-s + 0.497·24-s − 0.582·25-s + 0.264·26-s − 1.08·27-s + 1.61·28-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 \( 1 + 0.928T + 8T^{2} \)
3 \( 1 - 4.15T + 27T^{2} \)
5 \( 1 + 7.22T + 125T^{2} \)
7 \( 1 + 33.4T + 343T^{2} \)
11 \( 1 - 21.8T + 1.33e3T^{2} \)
13 \( 1 + 37.7T + 2.19e3T^{2} \)
17 \( 1 - 87.4T + 4.91e3T^{2} \)
19 \( 1 - 115.T + 6.85e3T^{2} \)
23 \( 1 + 3.50T + 1.21e4T^{2} \)
29 \( 1 - 149.T + 2.43e4T^{2} \)
31 \( 1 - 53.4T + 2.97e4T^{2} \)
37 \( 1 - 272.T + 5.06e4T^{2} \)
41 \( 1 + 446.T + 6.89e4T^{2} \)
47 \( 1 - 526.T + 1.03e5T^{2} \)
53 \( 1 + 262.T + 1.48e5T^{2} \)
59 \( 1 - 575.T + 2.05e5T^{2} \)
61 \( 1 - 112.T + 2.26e5T^{2} \)
67 \( 1 + 6.38T + 3.00e5T^{2} \)
71 \( 1 + 570.T + 3.57e5T^{2} \)
73 \( 1 + 424.T + 3.89e5T^{2} \)
79 \( 1 + 395.T + 4.93e5T^{2} \)
83 \( 1 - 472.T + 5.71e5T^{2} \)
89 \( 1 - 836.T + 7.04e5T^{2} \)
97 \( 1 + 640.T + 9.12e5T^{2} \)
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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

−8.607169083646458367712958488620, −7.78082528313627556433322874195, −7.23703058771225503047946935928, −6.10870867757101615737138028488, −5.23992987465138179787212690795, −4.00264860097396840796037091447, −3.41182125702676138740521387882, −2.75515027412120210450905481481, −0.959372897747489400776356758491, 0, 0.959372897747489400776356758491, 2.75515027412120210450905481481, 3.41182125702676138740521387882, 4.00264860097396840796037091447, 5.23992987465138179787212690795, 6.10870867757101615737138028488, 7.23703058771225503047946935928, 7.78082528313627556433322874195, 8.607169083646458367712958488620

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