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Properties

Label 1-43e2-1849.1248-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.591 + 0.806i$
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  + (−0.976 + 0.217i)2-s + (0.520 − 0.853i)3-s + (0.905 − 0.424i)4-s + (−0.872 − 0.489i)5-s + (−0.322 + 0.946i)6-s + (0.744 + 0.667i)7-s + (−0.791 + 0.611i)8-s + (−0.457 − 0.889i)9-s + (0.957 + 0.288i)10-s + (−0.581 − 0.813i)11-s + (0.109 − 0.994i)12-s + (−0.322 − 0.946i)13-s + (−0.872 − 0.489i)14-s + (−0.872 + 0.489i)15-s + (0.639 − 0.768i)16-s + (−0.0365 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.217i)2-s + (0.520 − 0.853i)3-s + (0.905 − 0.424i)4-s + (−0.872 − 0.489i)5-s + (−0.322 + 0.946i)6-s + (0.744 + 0.667i)7-s + (−0.791 + 0.611i)8-s + (−0.457 − 0.889i)9-s + (0.957 + 0.288i)10-s + (−0.581 − 0.813i)11-s + (0.109 − 0.994i)12-s + (−0.322 − 0.946i)13-s + (−0.872 − 0.489i)14-s + (−0.872 + 0.489i)15-s + (0.639 − 0.768i)16-s + (−0.0365 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.591 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5384649773 + 0.2728076636i\)
\(L(\frac12)\) \(\approx\) \(0.5384649773 + 0.2728076636i\)
\(L(1)\) \(\approx\) \(0.6536826519 - 0.1023736508i\)
\(L(1)\) \(\approx\) \(0.6536826519 - 0.1023736508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.976 + 0.217i)T \)
3 \( 1 + (0.520 - 0.853i)T \)
5 \( 1 + (-0.872 - 0.489i)T \)
7 \( 1 + (0.744 + 0.667i)T \)
11 \( 1 + (-0.581 - 0.813i)T \)
13 \( 1 + (-0.322 - 0.946i)T \)
17 \( 1 + (-0.0365 + 0.999i)T \)
19 \( 1 + T \)
23 \( 1 + (0.109 + 0.994i)T \)
29 \( 1 + (-0.457 + 0.889i)T \)
31 \( 1 + (-0.322 + 0.946i)T \)
37 \( 1 + (-0.694 + 0.719i)T \)
41 \( 1 + (-0.457 - 0.889i)T \)
47 \( 1 + (-0.457 + 0.889i)T \)
53 \( 1 + (-0.791 - 0.611i)T \)
59 \( 1 + (0.252 + 0.967i)T \)
61 \( 1 + (0.744 + 0.667i)T \)
67 \( 1 + (-0.976 + 0.217i)T \)
71 \( 1 + (-0.457 - 0.889i)T \)
73 \( 1 + (0.905 + 0.424i)T \)
79 \( 1 + (0.833 + 0.551i)T \)
83 \( 1 + (0.744 + 0.667i)T \)
89 \( 1 + (-0.934 - 0.357i)T \)
97 \( 1 + (-0.322 + 0.946i)T \)
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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

−20.23871325380345240440355157959, −19.3116160019953287104369164965, −18.61158802568452510077162027752, −17.970403918940377223145338698853, −16.98509387645511286160014397889, −16.356796866422171268384936070250, −15.736621116242133880087705139785, −14.97587778379078039296894147345, −14.42282426649391802677890375449, −13.50805127155622787480856850603, −12.219231446789777044888649440818, −11.41369907907363126857487369721, −11.06465165218083478160226346090, −10.10436593592124703904415381987, −9.64372237182057848608986455951, −8.72889883540325060699493458524, −7.786520754195178371519466116631, −7.51941953342383421288688365590, −6.685297236236577679988163714607, −5.09303739273776145760239805828, −4.362374476400559556360440274028, −3.57003203303654674946039409755, −2.6167605942699578841508158109, −1.89707667451515326996294039612, −0.29914987814263483714368331971, 1.06205464473259612802774966620, 1.68196970869319886738564656377, 2.91020493353761719520710401525, 3.46617805329156339004041075892, 5.271244865204140716121395382034, 5.59585162311661055215273400337, 6.86484053009351386083187803695, 7.67994339588227060199544745442, 8.11230929935440397332232208645, 8.64786238415212241471992288896, 9.344221552953032268849749313257, 10.58520182169796819274523115897, 11.28270479462327206181116217818, 12.05121749296249326589849383124, 12.5802819761459490562778536166, 13.58483366834672095990386256531, 14.609138586230817032126462523572, 15.20851987190624967747326021202, 15.77476849251161499692337140220, 16.634447368638070562999971024882, 17.70976948268231348112103477328, 17.9174532089985339631923824962, 18.94272896595321082104879309082, 19.273698834072650406916046310251, 20.05374535907437883686939752207

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