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  + 1.84·2-s − 3.08·3-s − 4.57·4-s − 18.1·5-s − 5.70·6-s + 24.2·7-s − 23.2·8-s − 17.4·9-s − 33.6·10-s + 49.5·11-s + 14.1·12-s − 23.1·13-s + 44.7·14-s + 56.1·15-s − 6.40·16-s − 2.27·17-s − 32.3·18-s − 103.·19-s + 83.2·20-s − 74.7·21-s + 91.5·22-s − 159.·23-s + 71.8·24-s + 205.·25-s − 42.8·26-s + 137.·27-s − 110.·28-s + ⋯
L(s)  = 1  + 0.653·2-s − 0.593·3-s − 0.572·4-s − 1.62·5-s − 0.388·6-s + 1.30·7-s − 1.02·8-s − 0.647·9-s − 1.06·10-s + 1.35·11-s + 0.339·12-s − 0.494·13-s + 0.854·14-s + 0.966·15-s − 0.100·16-s − 0.0324·17-s − 0.423·18-s − 1.25·19-s + 0.931·20-s − 0.776·21-s + 0.887·22-s − 1.44·23-s + 0.610·24-s + 1.64·25-s − 0.323·26-s + 0.978·27-s − 0.748·28-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 \( 1 - 1.84T + 8T^{2} \)
3 \( 1 + 3.08T + 27T^{2} \)
5 \( 1 + 18.1T + 125T^{2} \)
7 \( 1 - 24.2T + 343T^{2} \)
11 \( 1 - 49.5T + 1.33e3T^{2} \)
13 \( 1 + 23.1T + 2.19e3T^{2} \)
17 \( 1 + 2.27T + 4.91e3T^{2} \)
19 \( 1 + 103.T + 6.85e3T^{2} \)
23 \( 1 + 159.T + 1.21e4T^{2} \)
29 \( 1 - 309.T + 2.43e4T^{2} \)
31 \( 1 - 93.3T + 2.97e4T^{2} \)
37 \( 1 - 1.12T + 5.06e4T^{2} \)
41 \( 1 - 386.T + 6.89e4T^{2} \)
47 \( 1 - 214.T + 1.03e5T^{2} \)
53 \( 1 + 96.4T + 1.48e5T^{2} \)
59 \( 1 + 135.T + 2.05e5T^{2} \)
61 \( 1 - 374.T + 2.26e5T^{2} \)
67 \( 1 - 266.T + 3.00e5T^{2} \)
71 \( 1 + 438.T + 3.57e5T^{2} \)
73 \( 1 - 809.T + 3.89e5T^{2} \)
79 \( 1 + 1.05e3T + 4.93e5T^{2} \)
83 \( 1 + 429.T + 5.71e5T^{2} \)
89 \( 1 - 136.T + 7.04e5T^{2} \)
97 \( 1 - 25.6T + 9.12e5T^{2} \)
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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

−8.356581948482559758782053457220, −7.945846456144178939282605206329, −6.72580760248832049491320704660, −5.99762185343508407642072185386, −4.92915136768801597014678538590, −4.31363341452083467833280920715, −3.95836030603199930596113881961, −2.64707489447273031032666071739, −0.970975010203764106289884219143, 0, 0.970975010203764106289884219143, 2.64707489447273031032666071739, 3.95836030603199930596113881961, 4.31363341452083467833280920715, 4.92915136768801597014678538590, 5.99762185343508407642072185386, 6.72580760248832049491320704660, 7.945846456144178939282605206329, 8.356581948482559758782053457220

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