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  + 4.32·2-s − 4.31·3-s + 10.7·4-s + 7.58·5-s − 18.6·6-s + 15.1·7-s + 11.7·8-s − 8.36·9-s + 32.8·10-s − 4.61·11-s − 46.2·12-s + 4.43·13-s + 65.5·14-s − 32.7·15-s − 34.8·16-s − 58.9·17-s − 36.1·18-s − 40.0·19-s + 81.3·20-s − 65.4·21-s − 19.9·22-s + 205.·23-s − 50.8·24-s − 67.4·25-s + 19.2·26-s + 152.·27-s + 162.·28-s + ⋯
L(s)  = 1  + 1.52·2-s − 0.830·3-s + 1.34·4-s + 0.678·5-s − 1.27·6-s + 0.818·7-s + 0.520·8-s − 0.309·9-s + 1.03·10-s − 0.126·11-s − 1.11·12-s + 0.0946·13-s + 1.25·14-s − 0.563·15-s − 0.543·16-s − 0.840·17-s − 0.473·18-s − 0.483·19-s + 0.909·20-s − 0.679·21-s − 0.193·22-s + 1.86·23-s − 0.432·24-s − 0.539·25-s + 0.144·26-s + 1.08·27-s + 1.09·28-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 \( 1 - 4.32T + 8T^{2} \)
3 \( 1 + 4.31T + 27T^{2} \)
5 \( 1 - 7.58T + 125T^{2} \)
7 \( 1 - 15.1T + 343T^{2} \)
11 \( 1 + 4.61T + 1.33e3T^{2} \)
13 \( 1 - 4.43T + 2.19e3T^{2} \)
17 \( 1 + 58.9T + 4.91e3T^{2} \)
19 \( 1 + 40.0T + 6.85e3T^{2} \)
23 \( 1 - 205.T + 1.21e4T^{2} \)
29 \( 1 + 248.T + 2.43e4T^{2} \)
31 \( 1 + 63.1T + 2.97e4T^{2} \)
37 \( 1 + 151.T + 5.06e4T^{2} \)
41 \( 1 + 292.T + 6.89e4T^{2} \)
47 \( 1 + 100.T + 1.03e5T^{2} \)
53 \( 1 - 566.T + 1.48e5T^{2} \)
59 \( 1 - 730.T + 2.05e5T^{2} \)
61 \( 1 + 928.T + 2.26e5T^{2} \)
67 \( 1 + 61.7T + 3.00e5T^{2} \)
71 \( 1 + 66.4T + 3.57e5T^{2} \)
73 \( 1 + 408.T + 3.89e5T^{2} \)
79 \( 1 - 583.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 - 977.T + 7.04e5T^{2} \)
97 \( 1 - 254.T + 9.12e5T^{2} \)
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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

−8.560904057758900012705723175243, −7.24411364995696400961334670395, −6.56684351495710456663695353896, −5.74596865167935478271172909864, −5.25011352622890158082707532706, −4.67386076323214094528831618670, −3.64212142749308450771570174161, −2.55503271242227587398342776515, −1.63362992202765739940701682516, 0, 1.63362992202765739940701682516, 2.55503271242227587398342776515, 3.64212142749308450771570174161, 4.67386076323214094528831618670, 5.25011352622890158082707532706, 5.74596865167935478271172909864, 6.56684351495710456663695353896, 7.24411364995696400961334670395, 8.560904057758900012705723175243

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