Properties

Label 1-43e2-1849.947-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.199 - 0.979i$
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  + (−0.322 − 0.946i)2-s + (−0.0365 + 0.999i)3-s + (−0.791 + 0.611i)4-s + (−0.694 + 0.719i)5-s + (0.957 − 0.288i)6-s + (−0.457 − 0.889i)7-s + (0.833 + 0.551i)8-s + (−0.997 − 0.0729i)9-s + (0.905 + 0.424i)10-s + (0.989 − 0.145i)11-s + (−0.581 − 0.813i)12-s + (0.957 + 0.288i)13-s + (−0.694 + 0.719i)14-s + (−0.694 − 0.719i)15-s + (0.252 − 0.967i)16-s + (0.744 − 0.667i)17-s + ⋯
L(s)  = 1  + (−0.322 − 0.946i)2-s + (−0.0365 + 0.999i)3-s + (−0.791 + 0.611i)4-s + (−0.694 + 0.719i)5-s + (0.957 − 0.288i)6-s + (−0.457 − 0.889i)7-s + (0.833 + 0.551i)8-s + (−0.997 − 0.0729i)9-s + (0.905 + 0.424i)10-s + (0.989 − 0.145i)11-s + (−0.581 − 0.813i)12-s + (0.957 + 0.288i)13-s + (−0.694 + 0.719i)14-s + (−0.694 − 0.719i)15-s + (0.252 − 0.967i)16-s + (0.744 − 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.199 - 0.979i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.199 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6203948452 - 0.5065592359i\)
\(L(\frac12)\) \(\approx\) \(0.6203948452 - 0.5065592359i\)
\(L(1)\) \(\approx\) \(0.7159821815 - 0.1219316143i\)
\(L(1)\) \(\approx\) \(0.7159821815 - 0.1219316143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.322 - 0.946i)T \)
3 \( 1 + (-0.0365 + 0.999i)T \)
5 \( 1 + (-0.694 + 0.719i)T \)
7 \( 1 + (-0.457 - 0.889i)T \)
11 \( 1 + (0.989 - 0.145i)T \)
13 \( 1 + (0.957 + 0.288i)T \)
17 \( 1 + (0.744 - 0.667i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.581 + 0.813i)T \)
29 \( 1 + (-0.997 + 0.0729i)T \)
31 \( 1 + (0.957 - 0.288i)T \)
37 \( 1 + (-0.934 - 0.357i)T \)
41 \( 1 + (-0.997 - 0.0729i)T \)
47 \( 1 + (-0.997 + 0.0729i)T \)
53 \( 1 + (0.833 - 0.551i)T \)
59 \( 1 + (0.391 - 0.920i)T \)
61 \( 1 + (-0.457 - 0.889i)T \)
67 \( 1 + (-0.322 - 0.946i)T \)
71 \( 1 + (-0.997 - 0.0729i)T \)
73 \( 1 + (-0.791 - 0.611i)T \)
79 \( 1 + (0.639 + 0.768i)T \)
83 \( 1 + (-0.457 - 0.889i)T \)
89 \( 1 + (0.520 - 0.853i)T \)
97 \( 1 + (0.957 - 0.288i)T \)
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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

−19.959540015739856910708116804430, −19.32014852281276935203946287442, −18.75805717117495849105223614254, −18.14348439838569867022633465417, −17.32013056840257573952688602145, −16.56495206338154305079672470385, −16.06491762561040954774510619776, −15.15616033495040363780933885675, −14.56227863943573449928307787307, −13.55121133290627931802279786490, −13.02631670373016025366034594516, −12.016558867505142975092142756360, −11.811880904077484246545165011121, −10.41690712987567558419831140062, −9.36759885046013096849231227528, −8.6022747376996591999863082374, −8.300762005248660165836749246925, −7.37971025397475598737303535593, −6.57017037271627293003661459141, −5.86069660085942681939391637048, −5.27758998582090647973583389668, −4.04734843089266787165293864499, −3.19655155571044973423851504286, −1.61069532232936884977247683741, −0.99656801173859791391530562739, 0.40887935548484260517085136671, 1.609034043785974364359441068194, 3.17138656813816385616828214561, 3.48666832468141021762765915227, 4.002263177538950317785770575040, 4.979965903346997365578988856674, 6.158443683565298547116623696283, 7.20189260909003337128738822074, 7.98153730236222244347201136758, 8.92072095338583951672032421697, 9.71881121999868658311898434765, 10.16252202080840276454776162999, 11.06319245730159820090017113450, 11.562876694812984782624067901026, 12.058984578308858916835767391370, 13.51912967786475151867494333318, 13.95220785691713682874412562085, 14.65559633323821165935869244692, 15.81465755745710781165244089142, 16.30138278814957928258007349406, 17.030644354972623241868863801420, 17.832938120932730474279995457667, 18.732873426550124391651001341380, 19.37835098011368966869632325833, 20.06832206663880260264706189900

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