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  − 3.90·2-s + 3.52·3-s + 7.24·4-s − 10.7·5-s − 13.7·6-s + 28.1·7-s + 2.93·8-s − 14.6·9-s + 41.8·10-s + 63.0·11-s + 25.5·12-s − 23.3·13-s − 109.·14-s − 37.6·15-s − 69.4·16-s + 49.0·17-s + 57.0·18-s − 57.3·19-s − 77.6·20-s + 99.0·21-s − 246.·22-s + 137.·23-s + 10.3·24-s − 10.3·25-s + 91.2·26-s − 146.·27-s + 204.·28-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.677·3-s + 0.906·4-s − 0.957·5-s − 0.935·6-s + 1.51·7-s + 0.129·8-s − 0.541·9-s + 1.32·10-s + 1.72·11-s + 0.613·12-s − 0.498·13-s − 2.09·14-s − 0.648·15-s − 1.08·16-s + 0.699·17-s + 0.746·18-s − 0.691·19-s − 0.867·20-s + 1.02·21-s − 2.38·22-s + 1.25·23-s + 0.0878·24-s − 0.0831·25-s + 0.687·26-s − 1.04·27-s + 1.37·28-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 \( 1 + 3.90T + 8T^{2} \)
3 \( 1 - 3.52T + 27T^{2} \)
5 \( 1 + 10.7T + 125T^{2} \)
7 \( 1 - 28.1T + 343T^{2} \)
11 \( 1 - 63.0T + 1.33e3T^{2} \)
13 \( 1 + 23.3T + 2.19e3T^{2} \)
17 \( 1 - 49.0T + 4.91e3T^{2} \)
19 \( 1 + 57.3T + 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 + 129.T + 2.43e4T^{2} \)
31 \( 1 + 69.2T + 2.97e4T^{2} \)
37 \( 1 - 51.1T + 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
47 \( 1 + 432.T + 1.03e5T^{2} \)
53 \( 1 + 753.T + 1.48e5T^{2} \)
59 \( 1 + 845.T + 2.05e5T^{2} \)
61 \( 1 - 156.T + 2.26e5T^{2} \)
67 \( 1 - 618.T + 3.00e5T^{2} \)
71 \( 1 - 723.T + 3.57e5T^{2} \)
73 \( 1 - 315.T + 3.89e5T^{2} \)
79 \( 1 + 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 587.T + 5.71e5T^{2} \)
89 \( 1 - 1.07e3T + 7.04e5T^{2} \)
97 \( 1 + 402.T + 9.12e5T^{2} \)
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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

−8.446235558399557434880499059356, −7.959009635003590597294675087143, −7.43584748641185128591651146065, −6.51749450951263535175816714357, −5.08651236055751388649572177982, −4.25009359807869643925729035021, −3.33748403556263180539772212751, −1.94135218918626539615778470764, −1.26095590468163761996194840201, 0, 1.26095590468163761996194840201, 1.94135218918626539615778470764, 3.33748403556263180539772212751, 4.25009359807869643925729035021, 5.08651236055751388649572177982, 6.51749450951263535175816714357, 7.43584748641185128591651146065, 7.959009635003590597294675087143, 8.446235558399557434880499059356

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