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  + 3.06·2-s + 2.01·3-s + 1.40·4-s + 11.9·5-s + 6.17·6-s + 10.6·7-s − 20.2·8-s − 22.9·9-s + 36.6·10-s + 65.4·11-s + 2.82·12-s − 61.8·13-s + 32.6·14-s + 24.0·15-s − 73.2·16-s − 22.8·17-s − 70.3·18-s − 36.6·19-s + 16.7·20-s + 21.4·21-s + 200.·22-s − 143.·23-s − 40.7·24-s + 17.4·25-s − 189.·26-s − 100.·27-s + 14.9·28-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.387·3-s + 0.175·4-s + 1.06·5-s + 0.420·6-s + 0.575·7-s − 0.893·8-s − 0.849·9-s + 1.15·10-s + 1.79·11-s + 0.0680·12-s − 1.32·13-s + 0.623·14-s + 0.413·15-s − 1.14·16-s − 0.325·17-s − 0.921·18-s − 0.442·19-s + 0.187·20-s + 0.222·21-s + 1.94·22-s − 1.29·23-s − 0.346·24-s + 0.139·25-s − 1.43·26-s − 0.716·27-s + 0.101·28-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 \( 1 - 3.06T + 8T^{2} \)
3 \( 1 - 2.01T + 27T^{2} \)
5 \( 1 - 11.9T + 125T^{2} \)
7 \( 1 - 10.6T + 343T^{2} \)
11 \( 1 - 65.4T + 1.33e3T^{2} \)
13 \( 1 + 61.8T + 2.19e3T^{2} \)
17 \( 1 + 22.8T + 4.91e3T^{2} \)
19 \( 1 + 36.6T + 6.85e3T^{2} \)
23 \( 1 + 143.T + 1.21e4T^{2} \)
29 \( 1 + 213.T + 2.43e4T^{2} \)
31 \( 1 + 307.T + 2.97e4T^{2} \)
37 \( 1 - 83.7T + 5.06e4T^{2} \)
41 \( 1 - 59.1T + 6.89e4T^{2} \)
47 \( 1 - 201.T + 1.03e5T^{2} \)
53 \( 1 - 406.T + 1.48e5T^{2} \)
59 \( 1 - 164.T + 2.05e5T^{2} \)
61 \( 1 - 150.T + 2.26e5T^{2} \)
67 \( 1 - 169.T + 3.00e5T^{2} \)
71 \( 1 + 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 374.T + 3.89e5T^{2} \)
79 \( 1 - 465.T + 4.93e5T^{2} \)
83 \( 1 - 1.30e3T + 5.71e5T^{2} \)
89 \( 1 - 66.7T + 7.04e5T^{2} \)
97 \( 1 + 953.T + 9.12e5T^{2} \)
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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

−8.742555107676184112013564128270, −7.60974439196382429599691851325, −6.60082805413129371313885826426, −5.82718016229958793651220066743, −5.35086512517758784610264650135, −4.26111112597954368741870493350, −3.66436112051956711417787099118, −2.40056377808992424519156676176, −1.81792105813775485468766251179, 0, 1.81792105813775485468766251179, 2.40056377808992424519156676176, 3.66436112051956711417787099118, 4.26111112597954368741870493350, 5.35086512517758784610264650135, 5.82718016229958793651220066743, 6.60082805413129371313885826426, 7.60974439196382429599691851325, 8.742555107676184112013564128270

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