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  − 0.702·2-s + 4.82·3-s − 7.50·4-s + 8.51·5-s − 3.39·6-s + 25.5·7-s + 10.8·8-s − 3.71·9-s − 5.98·10-s − 28.9·11-s − 36.2·12-s − 77.8·13-s − 17.9·14-s + 41.0·15-s + 52.3·16-s + 87.6·17-s + 2.61·18-s + 15.5·19-s − 63.9·20-s + 123.·21-s + 20.3·22-s − 9.68·23-s + 52.5·24-s − 52.5·25-s + 54.7·26-s − 148.·27-s − 192.·28-s + ⋯
L(s)  = 1  − 0.248·2-s + 0.928·3-s − 0.938·4-s + 0.761·5-s − 0.230·6-s + 1.38·7-s + 0.481·8-s − 0.137·9-s − 0.189·10-s − 0.793·11-s − 0.871·12-s − 1.66·13-s − 0.343·14-s + 0.707·15-s + 0.818·16-s + 1.25·17-s + 0.0342·18-s + 0.187·19-s − 0.714·20-s + 1.28·21-s + 0.197·22-s − 0.0877·23-s + 0.447·24-s − 0.420·25-s + 0.412·26-s − 1.05·27-s − 1.29·28-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 \( 1 + 0.702T + 8T^{2} \)
3 \( 1 - 4.82T + 27T^{2} \)
5 \( 1 - 8.51T + 125T^{2} \)
7 \( 1 - 25.5T + 343T^{2} \)
11 \( 1 + 28.9T + 1.33e3T^{2} \)
13 \( 1 + 77.8T + 2.19e3T^{2} \)
17 \( 1 - 87.6T + 4.91e3T^{2} \)
19 \( 1 - 15.5T + 6.85e3T^{2} \)
23 \( 1 + 9.68T + 1.21e4T^{2} \)
29 \( 1 + 189.T + 2.43e4T^{2} \)
31 \( 1 - 260.T + 2.97e4T^{2} \)
37 \( 1 - 57.5T + 5.06e4T^{2} \)
41 \( 1 + 296.T + 6.89e4T^{2} \)
47 \( 1 + 360.T + 1.03e5T^{2} \)
53 \( 1 - 352.T + 1.48e5T^{2} \)
59 \( 1 + 711.T + 2.05e5T^{2} \)
61 \( 1 - 456.T + 2.26e5T^{2} \)
67 \( 1 + 488.T + 3.00e5T^{2} \)
71 \( 1 + 924.T + 3.57e5T^{2} \)
73 \( 1 + 708.T + 3.89e5T^{2} \)
79 \( 1 - 191.T + 4.93e5T^{2} \)
83 \( 1 - 962.T + 5.71e5T^{2} \)
89 \( 1 - 158.T + 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−8.433066914385885077562255059698, −7.82513928483429069219438311503, −7.48741521249418959289843657472, −5.76247502468821691078067606343, −5.16268807398993175829345872508, −4.53124130933371098616335823789, −3.26792776966760622669278759538, −2.32159107297291312343104942759, −1.46431854196156028475579271949, 0, 1.46431854196156028475579271949, 2.32159107297291312343104942759, 3.26792776966760622669278759538, 4.53124130933371098616335823789, 5.16268807398993175829345872508, 5.76247502468821691078067606343, 7.48741521249418959289843657472, 7.82513928483429069219438311503, 8.433066914385885077562255059698

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