Properties

Label 2-43e2-1.1-c3-0-430
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  + 5.33·2-s + 6.59·3-s + 20.5·4-s − 12.0·5-s + 35.2·6-s − 18.2·7-s + 66.8·8-s + 16.5·9-s − 64.4·10-s − 44.5·11-s + 135.·12-s − 45.4·13-s − 97.3·14-s − 79.6·15-s + 192.·16-s − 98.4·17-s + 88.1·18-s − 86.0·19-s − 247.·20-s − 120.·21-s − 237.·22-s + 142.·23-s + 440.·24-s + 20.8·25-s − 242.·26-s − 69.2·27-s − 373.·28-s + ⋯
L(s)  = 1  + 1.88·2-s + 1.26·3-s + 2.56·4-s − 1.08·5-s + 2.39·6-s − 0.984·7-s + 2.95·8-s + 0.611·9-s − 2.03·10-s − 1.22·11-s + 3.25·12-s − 0.969·13-s − 1.85·14-s − 1.37·15-s + 3.01·16-s − 1.40·17-s + 1.15·18-s − 1.03·19-s − 2.76·20-s − 1.24·21-s − 2.30·22-s + 1.29·23-s + 3.74·24-s + 0.166·25-s − 1.83·26-s − 0.493·27-s − 2.52·28-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 \( 1 - 5.33T + 8T^{2} \)
3 \( 1 - 6.59T + 27T^{2} \)
5 \( 1 + 12.0T + 125T^{2} \)
7 \( 1 + 18.2T + 343T^{2} \)
11 \( 1 + 44.5T + 1.33e3T^{2} \)
13 \( 1 + 45.4T + 2.19e3T^{2} \)
17 \( 1 + 98.4T + 4.91e3T^{2} \)
19 \( 1 + 86.0T + 6.85e3T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 - 162.T + 2.43e4T^{2} \)
31 \( 1 - 68.1T + 2.97e4T^{2} \)
37 \( 1 + 13.6T + 5.06e4T^{2} \)
41 \( 1 - 17.3T + 6.89e4T^{2} \)
47 \( 1 - 409.T + 1.03e5T^{2} \)
53 \( 1 + 161.T + 1.48e5T^{2} \)
59 \( 1 + 400.T + 2.05e5T^{2} \)
61 \( 1 + 706.T + 2.26e5T^{2} \)
67 \( 1 - 334.T + 3.00e5T^{2} \)
71 \( 1 - 480.T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + 250.T + 4.93e5T^{2} \)
83 \( 1 - 351.T + 5.71e5T^{2} \)
89 \( 1 + 117.T + 7.04e5T^{2} \)
97 \( 1 + 609.T + 9.12e5T^{2} \)
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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

−8.238947634279012732059019679816, −7.46185651815563519601966168493, −6.90980520199659170477671994255, −6.02863745184823709003527684712, −4.78112406687328918023496456493, −4.35352290485521904412448907285, −3.35115134683885930938649563608, −2.80018995284851445899769535569, −2.24606628322871973689621728452, 0, 2.24606628322871973689621728452, 2.80018995284851445899769535569, 3.35115134683885930938649563608, 4.35352290485521904412448907285, 4.78112406687328918023496456493, 6.02863745184823709003527684712, 6.90980520199659170477671994255, 7.46185651815563519601966168493, 8.238947634279012732059019679816

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