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  − 3.88·2-s + 9.09·3-s + 7.12·4-s − 19.0·5-s − 35.3·6-s + 18.7·7-s + 3.41·8-s + 55.7·9-s + 74.0·10-s + 4.37·11-s + 64.7·12-s + 45.6·13-s − 73.0·14-s − 173.·15-s − 70.2·16-s − 62.8·17-s − 216.·18-s + 33.1·19-s − 135.·20-s + 170.·21-s − 17.0·22-s − 32.0·23-s + 31.0·24-s + 237.·25-s − 177.·26-s + 261.·27-s + 133.·28-s + ⋯
L(s)  = 1  − 1.37·2-s + 1.75·3-s + 0.890·4-s − 1.70·5-s − 2.40·6-s + 1.01·7-s + 0.150·8-s + 2.06·9-s + 2.34·10-s + 0.119·11-s + 1.55·12-s + 0.973·13-s − 1.39·14-s − 2.98·15-s − 1.09·16-s − 0.896·17-s − 2.83·18-s + 0.399·19-s − 1.51·20-s + 1.77·21-s − 0.164·22-s − 0.290·23-s + 0.264·24-s + 1.90·25-s − 1.33·26-s + 1.86·27-s + 0.903·28-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 \( 1 + 3.88T + 8T^{2} \)
3 \( 1 - 9.09T + 27T^{2} \)
5 \( 1 + 19.0T + 125T^{2} \)
7 \( 1 - 18.7T + 343T^{2} \)
11 \( 1 - 4.37T + 1.33e3T^{2} \)
13 \( 1 - 45.6T + 2.19e3T^{2} \)
17 \( 1 + 62.8T + 4.91e3T^{2} \)
19 \( 1 - 33.1T + 6.85e3T^{2} \)
23 \( 1 + 32.0T + 1.21e4T^{2} \)
29 \( 1 + 291.T + 2.43e4T^{2} \)
31 \( 1 + 83.9T + 2.97e4T^{2} \)
37 \( 1 + 90.1T + 5.06e4T^{2} \)
41 \( 1 + 422.T + 6.89e4T^{2} \)
47 \( 1 + 363.T + 1.03e5T^{2} \)
53 \( 1 - 387.T + 1.48e5T^{2} \)
59 \( 1 + 248.T + 2.05e5T^{2} \)
61 \( 1 - 64.5T + 2.26e5T^{2} \)
67 \( 1 - 80.6T + 3.00e5T^{2} \)
71 \( 1 - 936.T + 3.57e5T^{2} \)
73 \( 1 + 651.T + 3.89e5T^{2} \)
79 \( 1 - 1.11e3T + 4.93e5T^{2} \)
83 \( 1 - 26.3T + 5.71e5T^{2} \)
89 \( 1 + 347.T + 7.04e5T^{2} \)
97 \( 1 - 291.T + 9.12e5T^{2} \)
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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

−8.391925891930729295421462301685, −8.080078253221397637040724289759, −7.44769986955650305496626463632, −6.86003433187500958468778203401, −4.87894539206862025277570541870, −3.96101518341460664293157922742, −3.46826231476750935325826906568, −2.09019492116343321049154455126, −1.34100561249662907126907315539, 0, 1.34100561249662907126907315539, 2.09019492116343321049154455126, 3.46826231476750935325826906568, 3.96101518341460664293157922742, 4.87894539206862025277570541870, 6.86003433187500958468778203401, 7.44769986955650305496626463632, 8.080078253221397637040724289759, 8.391925891930729295421462301685

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