↑

Properties

Label 1-43e2-1849.1162-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.570 + 0.821i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.457 − 0.889i)2-s + (0.391 − 0.920i)3-s + (−0.581 + 0.813i)4-s + (0.833 + 0.551i)5-s + (−0.997 + 0.0729i)6-s + (−0.872 + 0.489i)7-s + (0.989 + 0.145i)8-s + (−0.694 − 0.719i)9-s + (0.109 − 0.994i)10-s + (−0.0365 − 0.999i)11-s + (0.520 + 0.853i)12-s + (−0.997 − 0.0729i)13-s + (0.833 + 0.551i)14-s + (0.833 − 0.551i)15-s + (−0.322 − 0.946i)16-s + (−0.181 − 0.983i)17-s + ⋯
L(s)  = 1  + (−0.457 − 0.889i)2-s + (0.391 − 0.920i)3-s + (−0.581 + 0.813i)4-s + (0.833 + 0.551i)5-s + (−0.997 + 0.0729i)6-s + (−0.872 + 0.489i)7-s + (0.989 + 0.145i)8-s + (−0.694 − 0.719i)9-s + (0.109 − 0.994i)10-s + (−0.0365 − 0.999i)11-s + (0.520 + 0.853i)12-s + (−0.997 − 0.0729i)13-s + (0.833 + 0.551i)14-s + (0.833 − 0.551i)15-s + (−0.322 − 0.946i)16-s + (−0.181 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2167706365 - 0.4146668642i\)
\(L(\frac12)\) \(\approx\) \(-0.2167706365 - 0.4146668642i\)
\(L(1)\) \(\approx\) \(0.5839037587 - 0.4870350789i\)
\(L(1)\) \(\approx\) \(0.5839037587 - 0.4870350789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.457 - 0.889i)T \)
3 \( 1 + (0.391 - 0.920i)T \)
5 \( 1 + (0.833 + 0.551i)T \)
7 \( 1 + (-0.872 + 0.489i)T \)
11 \( 1 + (-0.0365 - 0.999i)T \)
13 \( 1 + (-0.997 - 0.0729i)T \)
17 \( 1 + (-0.181 - 0.983i)T \)
19 \( 1 + T \)
23 \( 1 + (0.520 - 0.853i)T \)
29 \( 1 + (-0.694 + 0.719i)T \)
31 \( 1 + (-0.997 + 0.0729i)T \)
37 \( 1 + (0.639 + 0.768i)T \)
41 \( 1 + (-0.694 - 0.719i)T \)
47 \( 1 + (-0.694 + 0.719i)T \)
53 \( 1 + (0.989 - 0.145i)T \)
59 \( 1 + (0.957 - 0.288i)T \)
61 \( 1 + (-0.872 + 0.489i)T \)
67 \( 1 + (-0.457 - 0.889i)T \)
71 \( 1 + (-0.694 - 0.719i)T \)
73 \( 1 + (-0.581 - 0.813i)T \)
79 \( 1 + (-0.976 - 0.217i)T \)
83 \( 1 + (-0.872 + 0.489i)T \)
89 \( 1 + (0.252 + 0.967i)T \)
97 \( 1 + (-0.997 + 0.0729i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.302736299959546959225591052863, −19.95314506277867251307244721081, −19.29899613105203574754079006016, −18.17557106430423573453961075223, −17.317445439942391588065214536588, −16.92386604765214244287280650483, −16.32012652566691229284999969197, −15.51423245078491336821334077381, −14.83040169985547172745686634392, −14.24378515456725656229885615594, −13.22746533873215019263800563468, −12.94308968091157394833436027535, −11.49584183129751001813598648194, −10.18458274153447107665937310983, −10.01105432425994327553013131458, −9.37340564242483968941732556623, −8.78108017282263501117915495126, −7.62951902257862221953880187878, −7.088387329104968874082147901805, −5.92056230331990269339744110785, −5.33907974795015156960311133566, −4.52559979606470217949119700909, −3.759815566267701949531303784894, −2.44953915992460961932554983769, −1.45462606153291284166288196077, 0.177877574373224716510330506837, 1.35097972847872093724685527758, 2.38041032457082197701635651172, 2.93288810376288656801327605762, 3.39715695468934147085410526029, 5.07446383984441085521137389969, 5.877280652967372491669280088971, 6.93882750642794116918150979411, 7.37704576934925990017530330499, 8.55109464740207917148650935832, 9.236359172605798013507485712189, 9.68365650237282432989978473511, 10.66806795908176617196503654161, 11.52108889568732688780217203724, 12.200782035363811818283337947080, 13.010427260940768694533075756010, 13.49663182967178850307911334245, 14.18756141551340077187237732132, 14.963524457603100496511160478298, 16.37309924140873982378982647240, 16.82948062001635920369624119750, 17.9366754040674981366904972195, 18.32260638258332391948376951500, 18.879937498415825946338840071240, 19.493642131720832870086844789505

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