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

−8.352931307904481706401586326913, −7.77774328519646859241785061046, −6.73450818840141616901091075156, −5.98587124638813488331458596960, −5.11312965420445524608514392786, −4.62750654936786405643272323680, −3.41022855313845556779944069678, −2.80871282916626533036279219192, −0.61272408772844635362821909163, 0, 0.61272408772844635362821909163, 2.80871282916626533036279219192, 3.41022855313845556779944069678, 4.62750654936786405643272323680, 5.11312965420445524608514392786, 5.98587124638813488331458596960, 6.73450818840141616901091075156, 7.77774328519646859241785061046, 8.352931307904481706401586326913

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