↑

Properties

Label 1-43e2-1849.775-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.337 - 0.941i$
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  + (0.744 + 0.667i)2-s + (0.252 + 0.967i)3-s + (0.109 + 0.994i)4-s + (−0.791 + 0.611i)5-s + (−0.457 + 0.889i)6-s + (−0.181 + 0.983i)7-s + (−0.581 + 0.813i)8-s + (−0.872 + 0.489i)9-s + (−0.997 − 0.0729i)10-s + (0.520 + 0.853i)11-s + (−0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.791 + 0.611i)14-s + (−0.791 − 0.611i)15-s + (−0.976 + 0.217i)16-s + (0.391 − 0.920i)17-s + ⋯
L(s)  = 1  + (0.744 + 0.667i)2-s + (0.252 + 0.967i)3-s + (0.109 + 0.994i)4-s + (−0.791 + 0.611i)5-s + (−0.457 + 0.889i)6-s + (−0.181 + 0.983i)7-s + (−0.581 + 0.813i)8-s + (−0.872 + 0.489i)9-s + (−0.997 − 0.0729i)10-s + (0.520 + 0.853i)11-s + (−0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.791 + 0.611i)14-s + (−0.791 − 0.611i)15-s + (−0.976 + 0.217i)16-s + (0.391 − 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8696210525 + 0.6121627195i\)
\(L(\frac12)\) \(\approx\) \(-0.8696210525 + 0.6121627195i\)
\(L(1)\) \(\approx\) \(0.5303994725 + 1.050876512i\)
\(L(1)\) \(\approx\) \(0.5303994725 + 1.050876512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.744 + 0.667i)T \)
3 \( 1 + (0.252 + 0.967i)T \)
5 \( 1 + (-0.791 + 0.611i)T \)
7 \( 1 + (-0.181 + 0.983i)T \)
11 \( 1 + (0.520 + 0.853i)T \)
13 \( 1 + (-0.457 - 0.889i)T \)
17 \( 1 + (0.391 - 0.920i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.934 - 0.357i)T \)
29 \( 1 + (-0.872 - 0.489i)T \)
31 \( 1 + (-0.457 + 0.889i)T \)
37 \( 1 + (0.833 + 0.551i)T \)
41 \( 1 + (-0.872 + 0.489i)T \)
47 \( 1 + (-0.872 - 0.489i)T \)
53 \( 1 + (-0.581 - 0.813i)T \)
59 \( 1 + (-0.322 + 0.946i)T \)
61 \( 1 + (-0.181 + 0.983i)T \)
67 \( 1 + (0.744 + 0.667i)T \)
71 \( 1 + (-0.872 + 0.489i)T \)
73 \( 1 + (0.109 - 0.994i)T \)
79 \( 1 + (0.989 + 0.145i)T \)
83 \( 1 + (-0.181 + 0.983i)T \)
89 \( 1 + (0.639 + 0.768i)T \)
97 \( 1 + (-0.457 + 0.889i)T \)
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

−19.83061360181413421774080662792, −18.99150666450619891841033396928, −18.55253646074167520129681085600, −17.24661485934933131465182667390, −16.629990194219672440161040068344, −15.85907150163142633972202973768, −14.68697144356834460909419965812, −14.20551497105532133402047876628, −13.51696323900216330568791263132, −12.84869161267145486152586031736, −12.16929668817282760494440611363, −11.476124258139812110768682904540, −11.00471501797738638759829342100, −9.69463068777455213802142272648, −9.09021012189590223168185782078, −7.961601937660147691874754918794, −7.371462953420579540144279582323, −6.434975405034518899908606374442, −5.70860983598067811042184638740, −4.62007556494022957981498039761, −3.62462131177897470055099005190, −3.43181814665993112095794520175, −1.905751259499131470872786564791, −1.26527686961396650830937040987, −0.2743962908474911128476603996, 2.29732659808013016588231032963, 3.05808832773447021657561985158, 3.59053399111459993681789821568, 4.60912197611904590672156067722, 5.2055563203076760866535874974, 6.02724282425432376422842563680, 7.05131815917719262738763320024, 7.792975201485661836031212437038, 8.45727136030866405477145166138, 9.479157879728653269394490294103, 10.06082515637687281817435426304, 11.32879685324113288401639738783, 11.850255090616025097288899127969, 12.42387217437220206376207871495, 13.558436053791152196612558445152, 14.490517806532469649344225549372, 14.92290326351238994058638000158, 15.37991625455048723581705252184, 16.13927871914093233577374672248, 16.59061406089016407310265859245, 17.84913124617098999893282245997, 18.24635163270048611913077687622, 19.49944954523426129348714161619, 20.215309418260284028834480574623, 20.694974251714506596063541510840

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