Properties

Label 1-43e2-1849.1033-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.129 - 0.991i$
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  + (−0.997 + 0.0729i)2-s + (−0.181 + 0.983i)3-s + (0.989 − 0.145i)4-s + (0.639 − 0.768i)5-s + (0.109 − 0.994i)6-s + (−0.694 + 0.719i)7-s + (−0.976 + 0.217i)8-s + (−0.934 − 0.357i)9-s + (−0.581 + 0.813i)10-s + (0.744 − 0.667i)11-s + (−0.0365 + 0.999i)12-s + (0.109 + 0.994i)13-s + (0.639 − 0.768i)14-s + (0.639 + 0.768i)15-s + (0.957 − 0.288i)16-s + (−0.872 + 0.489i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0729i)2-s + (−0.181 + 0.983i)3-s + (0.989 − 0.145i)4-s + (0.639 − 0.768i)5-s + (0.109 − 0.994i)6-s + (−0.694 + 0.719i)7-s + (−0.976 + 0.217i)8-s + (−0.934 − 0.357i)9-s + (−0.581 + 0.813i)10-s + (0.744 − 0.667i)11-s + (−0.0365 + 0.999i)12-s + (0.109 + 0.994i)13-s + (0.639 − 0.768i)14-s + (0.639 + 0.768i)15-s + (0.957 − 0.288i)16-s + (−0.872 + 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3164978619 - 0.2778189325i\)
\(L(\frac12)\) \(\approx\) \(0.3164978619 - 0.2778189325i\)
\(L(1)\) \(\approx\) \(0.5857033614 + 0.09430219268i\)
\(L(1)\) \(\approx\) \(0.5857033614 + 0.09430219268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.997 + 0.0729i)T \)
3 \( 1 + (-0.181 + 0.983i)T \)
5 \( 1 + (0.639 - 0.768i)T \)
7 \( 1 + (-0.694 + 0.719i)T \)
11 \( 1 + (0.744 - 0.667i)T \)
13 \( 1 + (0.109 + 0.994i)T \)
17 \( 1 + (-0.872 + 0.489i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.0365 - 0.999i)T \)
29 \( 1 + (-0.934 + 0.357i)T \)
31 \( 1 + (0.109 - 0.994i)T \)
37 \( 1 + (0.252 - 0.967i)T \)
41 \( 1 + (-0.934 - 0.357i)T \)
47 \( 1 + (-0.934 + 0.357i)T \)
53 \( 1 + (-0.976 - 0.217i)T \)
59 \( 1 + (0.905 + 0.424i)T \)
61 \( 1 + (-0.694 + 0.719i)T \)
67 \( 1 + (-0.997 + 0.0729i)T \)
71 \( 1 + (-0.934 - 0.357i)T \)
73 \( 1 + (0.989 + 0.145i)T \)
79 \( 1 + (-0.322 - 0.946i)T \)
83 \( 1 + (-0.694 + 0.719i)T \)
89 \( 1 + (0.391 + 0.920i)T \)
97 \( 1 + (0.109 - 0.994i)T \)
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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

−20.015276624009057647530675899829, −19.578653694183259228679582300193, −18.64091566972805776175449723478, −18.11566712919760255495425417620, −17.40701337783344519120639753617, −17.14883911699563621850274161435, −16.066710545741762753638393680190, −15.24594026288491684157631594486, −14.35579686699472517746451397054, −13.461034811379226273274953641874, −12.974424543452415383142546304074, −11.88033202942755568510892299886, −11.31895807623208783401682920494, −10.47422192755215988360910950227, −9.7544948724184143396976464734, −9.17350521231317784070927041833, −7.96593789812391619138683882010, −7.29973732058618707593037063647, −6.75307008091896542655746428033, −6.20542990532149699537714468089, −5.21626544969730756185915751658, −3.42521406680864053280448240213, −2.92332106183901470912135280383, −1.81816539488635956795353899778, −1.15358131796618116771475702752, 0.21694473481030267742044711068, 1.56799773261170835102303150719, 2.49095368763495044195412561555, 3.50564244235837193778822678971, 4.48516348155552355257401895300, 5.64091806217617208632260560555, 6.08537588498640442442971813423, 6.83396374742061525095414933945, 8.31257137500939644852145370039, 8.95176064281333833936028106781, 9.30004017764765808543292369838, 9.89303099722750955527147510221, 10.88007521882447398792138312143, 11.59191166273537621878237340285, 12.216732233895013000396337689390, 13.271866256875121773110756947678, 14.29251245988180825431840144557, 15.02920933767944497864818466346, 15.93433925274012314481131818062, 16.468799331534692768684404729891, 16.76546916471138734020650597974, 17.66689309567001115913594435083, 18.41369030599844013679095213453, 19.295492118318321475181988006532, 19.92487715351907746762392047716

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