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  + 0.0180·2-s + 4.14·3-s − 7.99·4-s − 13.3·5-s + 0.0747·6-s − 5.69·7-s − 0.288·8-s − 9.78·9-s − 0.240·10-s − 47.5·11-s − 33.1·12-s + 76.9·13-s − 0.102·14-s − 55.3·15-s + 63.9·16-s + 7.02·17-s − 0.176·18-s + 132.·19-s + 106.·20-s − 23.6·21-s − 0.855·22-s + 93.7·23-s − 1.19·24-s + 53.2·25-s + 1.38·26-s − 152.·27-s + 45.5·28-s + ⋯
L(s)  = 1  + 0.00636·2-s + 0.798·3-s − 0.999·4-s − 1.19·5-s + 0.00508·6-s − 0.307·7-s − 0.0127·8-s − 0.362·9-s − 0.00760·10-s − 1.30·11-s − 0.798·12-s + 1.64·13-s − 0.00195·14-s − 0.953·15-s + 0.999·16-s + 0.100·17-s − 0.00230·18-s + 1.59·19-s + 1.19·20-s − 0.245·21-s − 0.00829·22-s + 0.850·23-s − 0.0101·24-s + 0.425·25-s + 0.0104·26-s − 1.08·27-s + 0.307·28-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 \( 1 - 0.0180T + 8T^{2} \)
3 \( 1 - 4.14T + 27T^{2} \)
5 \( 1 + 13.3T + 125T^{2} \)
7 \( 1 + 5.69T + 343T^{2} \)
11 \( 1 + 47.5T + 1.33e3T^{2} \)
13 \( 1 - 76.9T + 2.19e3T^{2} \)
17 \( 1 - 7.02T + 4.91e3T^{2} \)
19 \( 1 - 132.T + 6.85e3T^{2} \)
23 \( 1 - 93.7T + 1.21e4T^{2} \)
29 \( 1 + 100.T + 2.43e4T^{2} \)
31 \( 1 + 169.T + 2.97e4T^{2} \)
37 \( 1 - 268.T + 5.06e4T^{2} \)
41 \( 1 - 157.T + 6.89e4T^{2} \)
47 \( 1 - 394.T + 1.03e5T^{2} \)
53 \( 1 - 447.T + 1.48e5T^{2} \)
59 \( 1 + 653.T + 2.05e5T^{2} \)
61 \( 1 - 168.T + 2.26e5T^{2} \)
67 \( 1 - 646.T + 3.00e5T^{2} \)
71 \( 1 - 61.1T + 3.57e5T^{2} \)
73 \( 1 + 336.T + 3.89e5T^{2} \)
79 \( 1 - 664.T + 4.93e5T^{2} \)
83 \( 1 + 242.T + 5.71e5T^{2} \)
89 \( 1 + 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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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

−8.451664772652355161928315735614, −7.85403607557078439540651576877, −7.37681066502977222960447046593, −5.84147768119325301717525904062, −5.24113405422683590856939323988, −4.04242111959961841495487032325, −3.49597433092303631499630996355, −2.80874944336240227850959514714, −1.03394619289072975829761352447, 0, 1.03394619289072975829761352447, 2.80874944336240227850959514714, 3.49597433092303631499630996355, 4.04242111959961841495487032325, 5.24113405422683590856939323988, 5.84147768119325301717525904062, 7.37681066502977222960447046593, 7.85403607557078439540651576877, 8.451664772652355161928315735614

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