Properties

Label 2-43e2-1.1-c3-0-296
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.54·2-s − 3.02·3-s + 4.56·4-s + 20.5·5-s + 10.7·6-s + 15.8·7-s + 12.1·8-s − 17.8·9-s − 72.9·10-s − 34.9·11-s − 13.8·12-s − 56.8·13-s − 56.1·14-s − 62.3·15-s − 79.6·16-s + 107.·17-s + 63.1·18-s + 82.7·19-s + 93.9·20-s − 47.9·21-s + 123.·22-s − 34.1·23-s − 36.9·24-s + 299.·25-s + 201.·26-s + 135.·27-s + 72.2·28-s + ⋯
L(s)  = 1  − 1.25·2-s − 0.582·3-s + 0.570·4-s + 1.84·5-s + 0.730·6-s + 0.855·7-s + 0.538·8-s − 0.660·9-s − 2.30·10-s − 0.957·11-s − 0.332·12-s − 1.21·13-s − 1.07·14-s − 1.07·15-s − 1.24·16-s + 1.53·17-s + 0.827·18-s + 0.998·19-s + 1.05·20-s − 0.498·21-s + 1.19·22-s − 0.309·23-s − 0.313·24-s + 2.39·25-s + 1.51·26-s + 0.967·27-s + 0.487·28-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: $-1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
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 \( 1 + 3.54T + 8T^{2} \)
3 \( 1 + 3.02T + 27T^{2} \)
5 \( 1 - 20.5T + 125T^{2} \)
7 \( 1 - 15.8T + 343T^{2} \)
11 \( 1 + 34.9T + 1.33e3T^{2} \)
13 \( 1 + 56.8T + 2.19e3T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 - 82.7T + 6.85e3T^{2} \)
23 \( 1 + 34.1T + 1.21e4T^{2} \)
29 \( 1 + 201.T + 2.43e4T^{2} \)
31 \( 1 + 43.9T + 2.97e4T^{2} \)
37 \( 1 + 170.T + 5.06e4T^{2} \)
41 \( 1 - 28.1T + 6.89e4T^{2} \)
47 \( 1 - 37.3T + 1.03e5T^{2} \)
53 \( 1 + 249.T + 1.48e5T^{2} \)
59 \( 1 - 8.65T + 2.05e5T^{2} \)
61 \( 1 + 133.T + 2.26e5T^{2} \)
67 \( 1 + 660.T + 3.00e5T^{2} \)
71 \( 1 - 573.T + 3.57e5T^{2} \)
73 \( 1 + 403.T + 3.89e5T^{2} \)
79 \( 1 - 104.T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 - 194.T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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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

−8.616033455277269270467414564686, −7.76111071561790006645049928709, −7.20659275611438542179203809399, −5.94185306866102635050543899607, −5.31270777039523502679930625547, −4.94985489371774095281155353853, −2.92474938997300243861610376272, −1.99049466896823500226898975792, −1.21673122522232478758474748754, 0, 1.21673122522232478758474748754, 1.99049466896823500226898975792, 2.92474938997300243861610376272, 4.94985489371774095281155353853, 5.31270777039523502679930625547, 5.94185306866102635050543899607, 7.20659275611438542179203809399, 7.76111071561790006645049928709, 8.616033455277269270467414564686

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