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  + 1.06·2-s + 7.81·3-s − 6.86·4-s + 6.14·5-s + 8.32·6-s − 5.24·7-s − 15.8·8-s + 34.1·9-s + 6.53·10-s − 14.5·11-s − 53.6·12-s − 49.9·13-s − 5.58·14-s + 48.0·15-s + 38.0·16-s + 75.6·17-s + 36.3·18-s + 128.·19-s − 42.1·20-s − 41.0·21-s − 15.4·22-s − 131.·23-s − 123.·24-s − 87.2·25-s − 53.1·26-s + 55.8·27-s + 36.0·28-s + ⋯
L(s)  = 1  + 0.376·2-s + 1.50·3-s − 0.858·4-s + 0.549·5-s + 0.566·6-s − 0.283·7-s − 0.699·8-s + 1.26·9-s + 0.206·10-s − 0.397·11-s − 1.29·12-s − 1.06·13-s − 0.106·14-s + 0.826·15-s + 0.595·16-s + 1.07·17-s + 0.475·18-s + 1.54·19-s − 0.471·20-s − 0.426·21-s − 0.149·22-s − 1.19·23-s − 1.05·24-s − 0.698·25-s − 0.400·26-s + 0.398·27-s + 0.243·28-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 \( 1 - 1.06T + 8T^{2} \)
3 \( 1 - 7.81T + 27T^{2} \)
5 \( 1 - 6.14T + 125T^{2} \)
7 \( 1 + 5.24T + 343T^{2} \)
11 \( 1 + 14.5T + 1.33e3T^{2} \)
13 \( 1 + 49.9T + 2.19e3T^{2} \)
17 \( 1 - 75.6T + 4.91e3T^{2} \)
19 \( 1 - 128.T + 6.85e3T^{2} \)
23 \( 1 + 131.T + 1.21e4T^{2} \)
29 \( 1 + 36.6T + 2.43e4T^{2} \)
31 \( 1 + 251.T + 2.97e4T^{2} \)
37 \( 1 + 264.T + 5.06e4T^{2} \)
41 \( 1 + 0.0646T + 6.89e4T^{2} \)
47 \( 1 - 346.T + 1.03e5T^{2} \)
53 \( 1 - 468.T + 1.48e5T^{2} \)
59 \( 1 - 586.T + 2.05e5T^{2} \)
61 \( 1 + 592.T + 2.26e5T^{2} \)
67 \( 1 + 472.T + 3.00e5T^{2} \)
71 \( 1 + 548.T + 3.57e5T^{2} \)
73 \( 1 + 328.T + 3.89e5T^{2} \)
79 \( 1 - 801.T + 4.93e5T^{2} \)
83 \( 1 + 1.34e3T + 5.71e5T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + 281.T + 9.12e5T^{2} \)
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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

−8.596980698645103843333439585859, −7.70972188275777345597830504334, −7.30462894667235504912241041119, −5.71914806250836620979169046363, −5.32711166745733490600671305952, −4.09125381310200553280050321947, −3.39386615940578585045966712293, −2.66019338081243540569924929759, −1.59342264801720836345663641566, 0, 1.59342264801720836345663641566, 2.66019338081243540569924929759, 3.39386615940578585045966712293, 4.09125381310200553280050321947, 5.32711166745733490600671305952, 5.71914806250836620979169046363, 7.30462894667235504912241041119, 7.70972188275777345597830504334, 8.596980698645103843333439585859

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