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Properties

Label 1-43e2-1849.1334-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.631 - 0.775i$
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  + (−0.0365 + 0.999i)2-s + (0.639 − 0.768i)3-s + (−0.997 − 0.0729i)4-s + (0.905 + 0.424i)5-s + (0.744 + 0.667i)6-s + (0.391 − 0.920i)7-s + (0.109 − 0.994i)8-s + (−0.181 − 0.983i)9-s + (−0.457 + 0.889i)10-s + (−0.934 − 0.357i)11-s + (−0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.905 + 0.424i)14-s + (0.905 − 0.424i)15-s + (0.989 + 0.145i)16-s + (0.252 − 0.967i)17-s + ⋯
L(s)  = 1  + (−0.0365 + 0.999i)2-s + (0.639 − 0.768i)3-s + (−0.997 − 0.0729i)4-s + (0.905 + 0.424i)5-s + (0.744 + 0.667i)6-s + (0.391 − 0.920i)7-s + (0.109 − 0.994i)8-s + (−0.181 − 0.983i)9-s + (−0.457 + 0.889i)10-s + (−0.934 − 0.357i)11-s + (−0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.905 + 0.424i)14-s + (0.905 − 0.424i)15-s + (0.989 + 0.145i)16-s + (0.252 − 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860297467 - 0.8836656002i\)
\(L(\frac12)\) \(\approx\) \(1.860297467 - 0.8836656002i\)
\(L(1)\) \(\approx\) \(1.371710772 - 0.03631160736i\)
\(L(1)\) \(\approx\) \(1.371710772 - 0.03631160736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.0365 + 0.999i)T \)
3 \( 1 + (0.639 - 0.768i)T \)
5 \( 1 + (0.905 + 0.424i)T \)
7 \( 1 + (0.391 - 0.920i)T \)
11 \( 1 + (-0.934 - 0.357i)T \)
13 \( 1 + (0.744 - 0.667i)T \)
17 \( 1 + (0.252 - 0.967i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.694 - 0.719i)T \)
29 \( 1 + (-0.181 + 0.983i)T \)
31 \( 1 + (0.744 + 0.667i)T \)
37 \( 1 + (-0.791 - 0.611i)T \)
41 \( 1 + (-0.181 - 0.983i)T \)
47 \( 1 + (-0.181 + 0.983i)T \)
53 \( 1 + (0.109 + 0.994i)T \)
59 \( 1 + (-0.976 + 0.217i)T \)
61 \( 1 + (0.391 - 0.920i)T \)
67 \( 1 + (-0.0365 + 0.999i)T \)
71 \( 1 + (-0.181 - 0.983i)T \)
73 \( 1 + (-0.997 + 0.0729i)T \)
79 \( 1 + (-0.581 - 0.813i)T \)
83 \( 1 + (0.391 - 0.920i)T \)
89 \( 1 + (0.833 - 0.551i)T \)
97 \( 1 + (0.744 + 0.667i)T \)
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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

−20.541655509737530897573417214805, −19.64899115278613220996135119484, −18.80055876791727179319820160529, −18.2286072078438059157181844914, −17.48107136333643433339316086099, −16.642399951297427913949287106153, −15.69897711537885877899248226571, −15.022347864234565200914577842837, −14.09556170446448934396480836824, −13.53354821018515330869219401763, −12.958429461344728004697992403, −11.89624155263931356557069375356, −11.30069205563029759159040463737, −10.16413119813136403088081770959, −9.900302381326047527950549693260, −9.09799475014790175499540309535, −8.39324453567420073957616794874, −7.877685311173085486323544113315, −6.03505876603973427146378794470, −5.35858021731340731001228512500, −4.70838419105027000224050818082, −3.791643569640064269365594352017, −2.846189019972503082658176066259, −2.066730461649854473664924750736, −1.50239278887778372473776377737, 0.67697766843576242489399125558, 1.54689911580185816952767033369, 2.90308640841901414209234276055, 3.487407060383932607504667809051, 4.82894758760427196679221350814, 5.62416138556873970463651399820, 6.35635291069823095699307553387, 7.27332855276233461791380521048, 7.62527657438484806939519646455, 8.51763538098454933546896490413, 9.234543120037379529098662716910, 10.22292903059982356405146475968, 10.71895400804304114306014639298, 12.12459006403779749429355152886, 13.10432910982654612132241962455, 13.579710015352380490999927644168, 14.1366727798444193038867020022, 14.52245537582399080489570709344, 15.73862995761344062476313354122, 16.19183955101748058702970592006, 17.41352045495021529395961968237, 17.77348680681305131264430176958, 18.446665720408535532882194259204, 18.84373611335333875123871036695, 20.15704000078898484584838415282

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