Properties

Label 2-43e2-1.1-c3-0-409
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  + 3.22·2-s + 8.71·3-s + 2.37·4-s − 13.5·5-s + 28.0·6-s + 24.0·7-s − 18.1·8-s + 49.0·9-s − 43.5·10-s − 71.6·11-s + 20.6·12-s + 26.7·13-s + 77.5·14-s − 117.·15-s − 77.3·16-s + 35.6·17-s + 157.·18-s − 88.4·19-s − 32.0·20-s + 209.·21-s − 230.·22-s − 64.6·23-s − 158.·24-s + 57.6·25-s + 86.2·26-s + 192.·27-s + 57.1·28-s + ⋯
L(s)  = 1  + 1.13·2-s + 1.67·3-s + 0.296·4-s − 1.20·5-s + 1.91·6-s + 1.29·7-s − 0.800·8-s + 1.81·9-s − 1.37·10-s − 1.96·11-s + 0.497·12-s + 0.571·13-s + 1.47·14-s − 2.02·15-s − 1.20·16-s + 0.508·17-s + 2.06·18-s − 1.06·19-s − 0.358·20-s + 2.18·21-s − 2.23·22-s − 0.586·23-s − 1.34·24-s + 0.461·25-s + 0.650·26-s + 1.36·27-s + 0.385·28-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: Trivial
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 \( 1 - 3.22T + 8T^{2} \)
3 \( 1 - 8.71T + 27T^{2} \)
5 \( 1 + 13.5T + 125T^{2} \)
7 \( 1 - 24.0T + 343T^{2} \)
11 \( 1 + 71.6T + 1.33e3T^{2} \)
13 \( 1 - 26.7T + 2.19e3T^{2} \)
17 \( 1 - 35.6T + 4.91e3T^{2} \)
19 \( 1 + 88.4T + 6.85e3T^{2} \)
23 \( 1 + 64.6T + 1.21e4T^{2} \)
29 \( 1 + 106.T + 2.43e4T^{2} \)
31 \( 1 + 7.27T + 2.97e4T^{2} \)
37 \( 1 - 195.T + 5.06e4T^{2} \)
41 \( 1 + 404.T + 6.89e4T^{2} \)
47 \( 1 - 31.1T + 1.03e5T^{2} \)
53 \( 1 - 34.4T + 1.48e5T^{2} \)
59 \( 1 + 486.T + 2.05e5T^{2} \)
61 \( 1 + 491.T + 2.26e5T^{2} \)
67 \( 1 - 102.T + 3.00e5T^{2} \)
71 \( 1 - 119.T + 3.57e5T^{2} \)
73 \( 1 + 123.T + 3.89e5T^{2} \)
79 \( 1 - 34.0T + 4.93e5T^{2} \)
83 \( 1 + 681.T + 5.71e5T^{2} \)
89 \( 1 + 1.56e3T + 7.04e5T^{2} \)
97 \( 1 - 93.1T + 9.12e5T^{2} \)
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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

−8.134790394892664721668084010630, −8.051721042882407720896301666926, −7.24143045870331519378061960324, −5.79993886166207043274328224484, −4.82623022721079908541928793993, −4.26149329520750580263781310869, −3.50729102261652822371173611896, −2.76355388019066949805908595814, −1.85344253317941680055920182006, 0, 1.85344253317941680055920182006, 2.76355388019066949805908595814, 3.50729102261652822371173611896, 4.26149329520750580263781310869, 4.82623022721079908541928793993, 5.79993886166207043274328224484, 7.24143045870331519378061960324, 8.051721042882407720896301666926, 8.134790394892664721668084010630

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