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Properties

Label 1-43e2-1849.44-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.967 - 0.252i$
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  + (−0.791 + 0.611i)2-s + (−0.997 − 0.0729i)3-s + (0.252 − 0.967i)4-s + (−0.0365 − 0.999i)5-s + (0.833 − 0.551i)6-s + (−0.581 + 0.813i)7-s + (0.391 + 0.920i)8-s + (0.989 + 0.145i)9-s + (0.639 + 0.768i)10-s + (0.957 − 0.288i)11-s + (−0.322 + 0.946i)12-s + (0.833 + 0.551i)13-s + (−0.0365 − 0.999i)14-s + (−0.0365 + 0.999i)15-s + (−0.872 − 0.489i)16-s + (0.109 − 0.994i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (−0.997 − 0.0729i)3-s + (0.252 − 0.967i)4-s + (−0.0365 − 0.999i)5-s + (0.833 − 0.551i)6-s + (−0.581 + 0.813i)7-s + (0.391 + 0.920i)8-s + (0.989 + 0.145i)9-s + (0.639 + 0.768i)10-s + (0.957 − 0.288i)11-s + (−0.322 + 0.946i)12-s + (0.833 + 0.551i)13-s + (−0.0365 − 0.999i)14-s + (−0.0365 + 0.999i)15-s + (−0.872 − 0.489i)16-s + (0.109 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.967 - 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8388749300 - 0.1074808682i\)
\(L(\frac12)\) \(\approx\) \(0.8388749300 - 0.1074808682i\)
\(L(1)\) \(\approx\) \(0.6321258377 + 0.02461749461i\)
\(L(1)\) \(\approx\) \(0.6321258377 + 0.02461749461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
3 \( 1 + (-0.997 - 0.0729i)T \)
5 \( 1 + (-0.0365 - 0.999i)T \)
7 \( 1 + (-0.581 + 0.813i)T \)
11 \( 1 + (0.957 - 0.288i)T \)
13 \( 1 + (0.833 + 0.551i)T \)
17 \( 1 + (0.109 - 0.994i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.322 - 0.946i)T \)
29 \( 1 + (0.989 - 0.145i)T \)
31 \( 1 + (0.833 - 0.551i)T \)
37 \( 1 + (0.744 + 0.667i)T \)
41 \( 1 + (0.989 + 0.145i)T \)
47 \( 1 + (0.989 - 0.145i)T \)
53 \( 1 + (0.391 - 0.920i)T \)
59 \( 1 + (-0.694 - 0.719i)T \)
61 \( 1 + (-0.581 + 0.813i)T \)
67 \( 1 + (-0.791 + 0.611i)T \)
71 \( 1 + (0.989 + 0.145i)T \)
73 \( 1 + (0.252 + 0.967i)T \)
79 \( 1 + (-0.181 + 0.983i)T \)
83 \( 1 + (-0.581 + 0.813i)T \)
89 \( 1 + (-0.457 - 0.889i)T \)
97 \( 1 + (0.833 - 0.551i)T \)
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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

−19.789103894053324330014619254519, −19.54743008127547956545786939573, −18.58429886315158987640251474437, −17.86844458337421616697453998877, −17.46798465982664310240926116577, −16.73205070429099137270305588986, −15.89657421254360396850398565154, −15.392921807358945592444205568999, −14.05063271718060952991367864984, −13.37883890019571895506600978302, −12.37192661275872867822236479227, −11.85716861970660758945365337785, −10.883595150465429753661279111533, −10.61676355105478422034191253034, −9.86735718790725508102975695326, −9.195156997019906270179285677530, −7.81424306919967425368587118844, −7.301039744180184815949237186306, −6.42227621814646784246295910360, −5.96896077863747406685490110975, −4.32386087482628226655392822948, −3.713821796612354812525536857900, −2.99610048174656565619854503324, −1.55562471653866517219497786376, −0.86430108510573463942175185478, 0.72012065111502389153683031049, 1.258499302978435248221022669855, 2.52718471690213630223734846812, 4.12071972199895754443312306826, 4.90344038572634222541744791355, 5.78161100478619461897112946507, 6.25755718472593697714739928712, 6.982069844662916556249494692470, 8.08283934348435385144239409302, 8.8444844265811642540722754230, 9.50192992121441978522792564401, 10.04615590378738308399205376082, 11.370719158980152444298221637449, 11.70420037756147242164708470902, 12.44052300683835041457041919475, 13.51014987989122886747929901958, 14.176785016817257408069801929004, 15.51335030069450406725614605326, 15.9161515313544218866250781524, 16.52281723523567411418886375960, 16.94305762518181381724711061418, 17.94673718241322368286232800591, 18.4342920545001618046059722725, 19.13296631843202719598060837346, 19.919032292532700495307982292611

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