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Properties

Label 1-43e2-1849.1291-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.414 + 0.910i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.639 − 0.768i)2-s + (−0.581 + 0.813i)3-s + (−0.181 − 0.983i)4-s + (−0.457 + 0.889i)5-s + (0.252 + 0.967i)6-s + (−0.976 + 0.217i)7-s + (−0.872 − 0.489i)8-s + (−0.322 − 0.946i)9-s + (0.391 + 0.920i)10-s + (−0.791 − 0.611i)11-s + (0.905 + 0.424i)12-s + (0.252 − 0.967i)13-s + (−0.457 + 0.889i)14-s + (−0.457 − 0.889i)15-s + (−0.934 + 0.357i)16-s + (0.989 + 0.145i)17-s + ⋯
L(s)  = 1  + (0.639 − 0.768i)2-s + (−0.581 + 0.813i)3-s + (−0.181 − 0.983i)4-s + (−0.457 + 0.889i)5-s + (0.252 + 0.967i)6-s + (−0.976 + 0.217i)7-s + (−0.872 − 0.489i)8-s + (−0.322 − 0.946i)9-s + (0.391 + 0.920i)10-s + (−0.791 − 0.611i)11-s + (0.905 + 0.424i)12-s + (0.252 − 0.967i)13-s + (−0.457 + 0.889i)14-s + (−0.457 − 0.889i)15-s + (−0.934 + 0.357i)16-s + (0.989 + 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.414 + 0.910i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.414 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1897299642 + 0.2948749631i\)
\(L(\frac12)\) \(\approx\) \(0.1897299642 + 0.2948749631i\)
\(L(1)\) \(\approx\) \(0.7949222565 - 0.1057652883i\)
\(L(1)\) \(\approx\) \(0.7949222565 - 0.1057652883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.639 - 0.768i)T \)
3 \( 1 + (-0.581 + 0.813i)T \)
5 \( 1 + (-0.457 + 0.889i)T \)
7 \( 1 + (-0.976 + 0.217i)T \)
11 \( 1 + (-0.791 - 0.611i)T \)
13 \( 1 + (0.252 - 0.967i)T \)
17 \( 1 + (0.989 + 0.145i)T \)
19 \( 1 + T \)
23 \( 1 + (0.905 - 0.424i)T \)
29 \( 1 + (-0.322 + 0.946i)T \)
31 \( 1 + (0.252 + 0.967i)T \)
37 \( 1 + (-0.997 + 0.0729i)T \)
41 \( 1 + (-0.322 - 0.946i)T \)
47 \( 1 + (-0.322 + 0.946i)T \)
53 \( 1 + (-0.872 + 0.489i)T \)
59 \( 1 + (0.520 - 0.853i)T \)
61 \( 1 + (-0.976 + 0.217i)T \)
67 \( 1 + (0.639 - 0.768i)T \)
71 \( 1 + (-0.322 - 0.946i)T \)
73 \( 1 + (-0.181 + 0.983i)T \)
79 \( 1 + (-0.694 + 0.719i)T \)
83 \( 1 + (-0.976 + 0.217i)T \)
89 \( 1 + (0.109 + 0.994i)T \)
97 \( 1 + (0.252 + 0.967i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.92451291162650220945724311930, −18.941060160398164533260550661495, −18.497222974680567020229233359489, −17.3160369110203257720736783124, −16.932796071476884543636607348434, −16.13239245177052736133804551097, −15.82043070621981821292550266802, −14.76201703328050017052446752271, −13.58664524976231212976464953913, −13.31714557870454525340124666153, −12.63150949323391678537222704558, −11.88440653663579120742877661747, −11.46973738806217739731180950561, −9.99951663359933536181764106457, −9.20366138141109493044284439200, −8.19161468483575711147113490388, −7.470561065020770015653973805123, −7.02038691675150549039881303291, −6.02609689638189928241869609360, −5.34985552564056533015169407741, −4.65927997498202531389035140289, −3.682381810692142031748232132, −2.73474470274687690633618685439, −1.43621677960645540309701110085, −0.12627767955120126425890401405, 1.03905368602353714568169261353, 2.91722491357407626240459441192, 3.14289271803531321084891930941, 3.71420187394193226532917078793, 5.0587409253417189524875420486, 5.5340382895728972844527444689, 6.31984348133717966403552217918, 7.17839998862248935698613524557, 8.46920553377736648805202422905, 9.46737707312394390066131725921, 10.22532786465876221394762117375, 10.67772008689855410237732385277, 11.23006075827886282642254203813, 12.29318180237347298585368517047, 12.605906529776764753158990358423, 13.757806339608430278653653971649, 14.43665527825599043119047632275, 15.35460840054871838562809430562, 15.73164491559769366433542404251, 16.36695179721858543704276140479, 17.602445768999841797892228944985, 18.47180857165388015099714298217, 18.86546051221948812736625278307, 19.736162561690587376801014569479, 20.533713103946037167529837643839

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