↑

Properties

Label 1-1157-1157.149-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.974 - 0.222i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.618 + 0.786i)2-s + (0.520 − 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (0.992 − 0.118i)6-s + (−0.771 + 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (−0.992 − 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯
L(s)  = 1  + (0.618 + 0.786i)2-s + (0.520 − 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (0.992 − 0.118i)6-s + (−0.771 + 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (−0.992 − 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.974 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.948099816 - 0.2194237493i\)
\(L(\frac12)\) \(\approx\) \(1.948099816 - 0.2194237493i\)
\(L(1)\) \(\approx\) \(1.422557860 + 0.1054164852i\)
\(L(1)\) \(\approx\) \(1.422557860 + 0.1054164852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
3 \( 1 + (0.520 - 0.853i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.771 + 0.636i)T \)
11 \( 1 + (0.0950 + 0.995i)T \)
17 \( 1 + (0.618 - 0.786i)T \)
19 \( 1 + (0.952 + 0.304i)T \)
23 \( 1 + (0.739 - 0.672i)T \)
29 \( 1 + (-0.165 - 0.986i)T \)
31 \( 1 + (0.977 + 0.212i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (0.999 - 0.0237i)T \)
43 \( 1 + (0.165 - 0.986i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.853 - 0.520i)T \)
61 \( 1 + (0.828 + 0.560i)T \)
67 \( 1 + (0.971 - 0.235i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.599 + 0.800i)T \)
97 \( 1 + (-0.0950 + 0.995i)T \)
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

−21.41701277257625002869158593646, −20.58214697875939295401536307574, −19.69848339762601160773221254933, −19.31538585165485040947731075345, −18.72579077520589557938603795997, −17.49113327201629192133533786999, −16.28937960604284885957446756243, −15.80717375334514616714732957302, −14.83150817989242995246370952930, −14.324081651413260266669323106955, −13.56684983842104745149872026374, −12.900301893448312557357129646599, −11.48407619925484885508903603719, −11.18619246431938931742922627711, −10.17464839491207603958566367428, −9.85024500529776978304961862070, −8.78898753086171516913844147877, −7.73212405500064522280493464838, −6.628319576941451337906374084687, −5.75716287727717050346560333568, −4.72812875839215020461626179555, −3.59870769656003425541887525073, −3.386100838254906180296227242284, −2.64292098488684099705299569228, −1.05842129901775556816052075265, 0.72921392935994768911085982839, 2.26298394285000824715327496251, 3.11497003055927515154142254052, 4.07287740683068520548718297269, 5.100010223845265269679050688820, 5.886286870074611702199724083936, 6.85256672259781152910968798966, 7.545101748637712641904602016616, 8.235153150354275504335050216806, 9.18905073111895850514516552541, 9.61188788676220074371954773155, 11.64879082573446166162173322538, 12.209452481297182996562424436791, 12.677489602024079183644814200147, 13.387662149665793831103026180818, 14.218908934660248365940254585211, 15.0598287954555049825801561811, 15.756659878027686053308454181134, 16.431312019244716608176135366711, 17.338353093211454545451753683418, 18.08344159286013633444472122280, 18.94164598912245884191588270572, 19.709590559932597269613662272904, 20.7342613019862315628028617869, 20.930961057868483756368285191941

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