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  + 2.34·2-s + 7.09·3-s − 2.49·4-s − 10.0·5-s + 16.6·6-s − 1.59·7-s − 24.6·8-s + 23.3·9-s − 23.5·10-s + 54.2·11-s − 17.7·12-s + 20.9·13-s − 3.73·14-s − 71.2·15-s − 37.8·16-s − 66.4·17-s + 54.8·18-s − 87.8·19-s + 25.0·20-s − 11.3·21-s + 127.·22-s + 106.·23-s − 174.·24-s − 24.3·25-s + 49.0·26-s − 25.6·27-s + 3.97·28-s + ⋯
L(s)  = 1  + 0.829·2-s + 1.36·3-s − 0.311·4-s − 0.897·5-s + 1.13·6-s − 0.0859·7-s − 1.08·8-s + 0.866·9-s − 0.744·10-s + 1.48·11-s − 0.426·12-s + 0.445·13-s − 0.0713·14-s − 1.22·15-s − 0.590·16-s − 0.947·17-s + 0.718·18-s − 1.06·19-s + 0.279·20-s − 0.117·21-s + 1.23·22-s + 0.969·23-s − 1.48·24-s − 0.194·25-s + 0.369·26-s − 0.183·27-s + 0.0268·28-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 \( 1 - 2.34T + 8T^{2} \)
3 \( 1 - 7.09T + 27T^{2} \)
5 \( 1 + 10.0T + 125T^{2} \)
7 \( 1 + 1.59T + 343T^{2} \)
11 \( 1 - 54.2T + 1.33e3T^{2} \)
13 \( 1 - 20.9T + 2.19e3T^{2} \)
17 \( 1 + 66.4T + 4.91e3T^{2} \)
19 \( 1 + 87.8T + 6.85e3T^{2} \)
23 \( 1 - 106.T + 1.21e4T^{2} \)
29 \( 1 - 270.T + 2.43e4T^{2} \)
31 \( 1 - 131.T + 2.97e4T^{2} \)
37 \( 1 + 62.8T + 5.06e4T^{2} \)
41 \( 1 + 464.T + 6.89e4T^{2} \)
47 \( 1 + 205.T + 1.03e5T^{2} \)
53 \( 1 + 296.T + 1.48e5T^{2} \)
59 \( 1 + 493.T + 2.05e5T^{2} \)
61 \( 1 + 617.T + 2.26e5T^{2} \)
67 \( 1 + 924.T + 3.00e5T^{2} \)
71 \( 1 + 254.T + 3.57e5T^{2} \)
73 \( 1 - 170.T + 3.89e5T^{2} \)
79 \( 1 + 234.T + 4.93e5T^{2} \)
83 \( 1 - 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + 1.51e3T + 7.04e5T^{2} \)
97 \( 1 - 273.T + 9.12e5T^{2} \)
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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

−8.642506583454014494727561668614, −7.965866820249055195954876099279, −6.71687639771787635562353947022, −6.30500975359807619067571101959, −4.69587658321276303442575812759, −4.28067702481598913453889591396, −3.45637839757850587671741794540, −2.91703287137278250060827656261, −1.54083760170176664150611376556, 0, 1.54083760170176664150611376556, 2.91703287137278250060827656261, 3.45637839757850587671741794540, 4.28067702481598913453889591396, 4.69587658321276303442575812759, 6.30500975359807619067571101959, 6.71687639771787635562353947022, 7.965866820249055195954876099279, 8.642506583454014494727561668614

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