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Properties

Label 1-43e2-1849.990-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.756 + 0.654i$
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  + (0.833 + 0.551i)2-s + (0.109 − 0.994i)3-s + (0.391 + 0.920i)4-s + (0.744 + 0.667i)5-s + (0.639 − 0.768i)6-s + (0.989 + 0.145i)7-s + (−0.181 + 0.983i)8-s + (−0.976 − 0.217i)9-s + (0.252 + 0.967i)10-s + (0.905 − 0.424i)11-s + (0.957 − 0.288i)12-s + (0.639 + 0.768i)13-s + (0.744 + 0.667i)14-s + (0.744 − 0.667i)15-s + (−0.694 + 0.719i)16-s + (−0.581 − 0.813i)17-s + ⋯
L(s)  = 1  + (0.833 + 0.551i)2-s + (0.109 − 0.994i)3-s + (0.391 + 0.920i)4-s + (0.744 + 0.667i)5-s + (0.639 − 0.768i)6-s + (0.989 + 0.145i)7-s + (−0.181 + 0.983i)8-s + (−0.976 − 0.217i)9-s + (0.252 + 0.967i)10-s + (0.905 − 0.424i)11-s + (0.957 − 0.288i)12-s + (0.639 + 0.768i)13-s + (0.744 + 0.667i)14-s + (0.744 − 0.667i)15-s + (−0.694 + 0.719i)16-s + (−0.581 − 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (990, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.661414721 + 1.364831864i\)
\(L(\frac12)\) \(\approx\) \(3.661414721 + 1.364831864i\)
\(L(1)\) \(\approx\) \(2.216658119 + 0.5026112747i\)
\(L(1)\) \(\approx\) \(2.216658119 + 0.5026112747i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.833 + 0.551i)T \)
3 \( 1 + (0.109 - 0.994i)T \)
5 \( 1 + (0.744 + 0.667i)T \)
7 \( 1 + (0.989 + 0.145i)T \)
11 \( 1 + (0.905 - 0.424i)T \)
13 \( 1 + (0.639 + 0.768i)T \)
17 \( 1 + (-0.581 - 0.813i)T \)
19 \( 1 + T \)
23 \( 1 + (0.957 + 0.288i)T \)
29 \( 1 + (-0.976 + 0.217i)T \)
31 \( 1 + (0.639 - 0.768i)T \)
37 \( 1 + (-0.457 - 0.889i)T \)
41 \( 1 + (-0.976 - 0.217i)T \)
47 \( 1 + (-0.976 + 0.217i)T \)
53 \( 1 + (-0.181 - 0.983i)T \)
59 \( 1 + (-0.934 + 0.357i)T \)
61 \( 1 + (0.989 + 0.145i)T \)
67 \( 1 + (0.833 + 0.551i)T \)
71 \( 1 + (-0.976 - 0.217i)T \)
73 \( 1 + (0.391 - 0.920i)T \)
79 \( 1 + (-0.872 + 0.489i)T \)
83 \( 1 + (0.989 + 0.145i)T \)
89 \( 1 + (-0.997 - 0.0729i)T \)
97 \( 1 + (0.639 - 0.768i)T \)
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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

−20.371662057960998964115051424306, −19.87197540537818676702569413926, −18.685852036142742559755930675787, −17.61676770540596528372202006492, −17.16882311637158677258367658571, −16.263868692410659474966029226395, −15.31642630603673249727046096765, −14.90889675402502289168351271680, −14.048004115640792431323773320, −13.54294602544975342234018990711, −12.66027239875098088792524593906, −11.75421038259236804308435264663, −11.10963911639927302367231081440, −10.38590870372648058452506882821, −9.70702407733628301690688521857, −8.90793725298173312473085876154, −8.21054951442112183272229041829, −6.77329188128166076354340170411, −5.86128655522263016171138344211, −5.121668538159845394705138983854, −4.64394816005058890338484333033, −3.80614858254181642630433437469, −2.95602521746032439048693121189, −1.78199096970473015080507033566, −1.15326972555380562289297588506, 1.36695323811998948827732761764, 2.05783540958646409620898925620, 2.97979786718876934207408373967, 3.81530145815836216430801550581, 5.070940488457373469806490786172, 5.68618655966668581401247883865, 6.57704590910826846966591693277, 7.007474597817799683743781004169, 7.79508472756242978328112982871, 8.78468003872018094485593340769, 9.318872721170871500873573883925, 11.05137708728195689941352148644, 11.43205263225216385314676060479, 11.953519859391250657126380697993, 13.26153366666668252728981726506, 13.55145757580483208034828888958, 14.345407662081060866056499960417, 14.599353068939613699854921433561, 15.62920610184807376491095061749, 16.70830426986299426442446342516, 17.29144437553379311176519477126, 17.98544669892952424436949311862, 18.5205667347845426691337838864, 19.394020478222541914479013858844, 20.53013215799008714277478503440

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