↑

Properties

Label 1-43e2-1849.259-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.318 + 0.948i$
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  + (−0.694 + 0.719i)2-s + (0.905 + 0.424i)3-s + (−0.0365 − 0.999i)4-s + (−0.976 + 0.217i)5-s + (−0.934 + 0.357i)6-s + (0.833 + 0.551i)7-s + (0.744 + 0.667i)8-s + (0.639 + 0.768i)9-s + (0.520 − 0.853i)10-s + (−0.181 − 0.983i)11-s + (0.391 − 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.976 + 0.217i)14-s + (−0.976 − 0.217i)15-s + (−0.997 + 0.0729i)16-s + (−0.791 − 0.611i)17-s + ⋯
L(s)  = 1  + (−0.694 + 0.719i)2-s + (0.905 + 0.424i)3-s + (−0.0365 − 0.999i)4-s + (−0.976 + 0.217i)5-s + (−0.934 + 0.357i)6-s + (0.833 + 0.551i)7-s + (0.744 + 0.667i)8-s + (0.639 + 0.768i)9-s + (0.520 − 0.853i)10-s + (−0.181 − 0.983i)11-s + (0.391 − 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.976 + 0.217i)14-s + (−0.976 − 0.217i)15-s + (−0.997 + 0.0729i)16-s + (−0.791 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.318 + 0.948i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.318 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098695891 + 0.7902906674i\)
\(L(\frac12)\) \(\approx\) \(1.098695891 + 0.7902906674i\)
\(L(1)\) \(\approx\) \(0.8621910298 + 0.4107323681i\)
\(L(1)\) \(\approx\) \(0.8621910298 + 0.4107323681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.694 + 0.719i)T \)
3 \( 1 + (0.905 + 0.424i)T \)
5 \( 1 + (-0.976 + 0.217i)T \)
7 \( 1 + (0.833 + 0.551i)T \)
11 \( 1 + (-0.181 - 0.983i)T \)
13 \( 1 + (-0.934 - 0.357i)T \)
17 \( 1 + (-0.791 - 0.611i)T \)
19 \( 1 + T \)
23 \( 1 + (0.391 + 0.920i)T \)
29 \( 1 + (0.639 - 0.768i)T \)
31 \( 1 + (-0.934 + 0.357i)T \)
37 \( 1 + (-0.322 - 0.946i)T \)
41 \( 1 + (0.639 + 0.768i)T \)
47 \( 1 + (0.639 - 0.768i)T \)
53 \( 1 + (0.744 - 0.667i)T \)
59 \( 1 + (0.109 - 0.994i)T \)
61 \( 1 + (0.833 + 0.551i)T \)
67 \( 1 + (-0.694 + 0.719i)T \)
71 \( 1 + (0.639 + 0.768i)T \)
73 \( 1 + (-0.0365 + 0.999i)T \)
79 \( 1 + (-0.457 - 0.889i)T \)
83 \( 1 + (0.833 + 0.551i)T \)
89 \( 1 + (0.957 + 0.288i)T \)
97 \( 1 + (-0.934 + 0.357i)T \)
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

−19.962759099580114125328546341046, −19.48403458811607482272149044103, −18.6078018744138035311976659161, −17.99134361085446987504449761203, −17.28380387020507473086829492101, −16.46838613680469001236871470641, −15.488910556753872164563290303223, −14.83020888135338833258663763603, −14.04823731634399760443254562252, −13.106746206901426007012512197187, −12.3481112865770460188638769029, −11.96182662128430252059074662187, −10.93330926340869261530687436503, −10.26951091818064499916658882475, −9.25974249947612751803076250242, −8.6973975633574637281568638600, −7.85860150183590610032439315545, −7.3516212131574809666305890365, −6.87740526568120289180298942034, −4.76754961603739794736532690028, −4.348974703913641140126823454882, −3.45801365515517194495849442114, −2.4772829948228915747192286898, −1.72983907098840411174371735892, −0.77716696739408338219863117529, 0.80810162604283038514259756907, 2.16413547924780252820154603788, 2.93993610241775520162888459238, 4.030474952951974259969528367849, 5.013684206284771656217550101879, 5.50138614459597593461971248724, 6.99501741385792046861710985296, 7.55270654491312253215339887109, 8.1614298496479576036899421164, 8.81895825440870591744139533124, 9.45616125209084232408529797584, 10.425155404845266447017231596130, 11.21753550679492429109730988869, 11.756663010388510095073247008234, 13.12614154767195751448327914375, 14.03711304846409615381856833724, 14.57830842639520292874634124192, 15.23007459870822629282823197211, 15.870747471169955367338367933113, 16.242149178530356531903280401977, 17.42061775097232701891372225099, 18.16750235775157524695935534488, 18.82482118874217106189609489718, 19.55242263934002029283045773473, 19.955938372974644107763328424023

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