↑

Properties

Label 1-1157-1157.119-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.542 - 0.840i$
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  + (0.814 − 0.580i)2-s + (−0.436 + 0.899i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (−0.281 − 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (−0.690 − 0.723i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (0.814 − 0.580i)2-s + (−0.436 + 0.899i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (−0.281 − 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (−0.690 − 0.723i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.542 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.436123187 - 1.327627546i\)
\(L(\frac12)\) \(\approx\) \(2.436123187 - 1.327627546i\)
\(L(1)\) \(\approx\) \(1.749955086 - 0.5074011490i\)
\(L(1)\) \(\approx\) \(1.749955086 - 0.5074011490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.814 - 0.580i)T \)
3 \( 1 + (-0.436 + 0.899i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.999 - 0.0237i)T \)
11 \( 1 + (-0.690 - 0.723i)T \)
17 \( 1 + (0.814 + 0.580i)T \)
19 \( 1 + (0.992 + 0.118i)T \)
23 \( 1 + (0.393 + 0.919i)T \)
29 \( 1 + (-0.520 - 0.853i)T \)
31 \( 1 + (-0.800 + 0.599i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.560 - 0.828i)T \)
43 \( 1 + (0.520 - 0.853i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.899 + 0.436i)T \)
61 \( 1 + (0.739 - 0.672i)T \)
67 \( 1 + (-0.945 + 0.327i)T \)
71 \( 1 + (-0.235 - 0.971i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (0.936 + 0.349i)T \)
97 \( 1 + (0.690 - 0.723i)T \)
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

−21.47811006647154804404612026345, −20.78472579078601968706753801479, −20.20813174698552332206059516942, −18.68977878791829233428135542169, −18.00857745184465127806426176125, −17.71401846122257200769859271345, −16.76821821412274737263472186109, −16.1240431788840709751494233812, −14.74551480361717479074843048246, −14.44205238989986285431652627005, −13.61140426706129908949078726924, −12.903517974110830423804347888166, −12.24947948489750390481166019806, −11.32707381348520448234031859239, −10.668182074352218301250009580787, −9.398883811795178474179431790949, −8.24748543449478964276748491090, −7.43505417012054995245205917380, −6.99370808356481238827127251081, −5.85833674218162802542465059141, −5.29092064061368048200822916926, −4.70197964670639824008003727780, −3.05427745319780539581126536271, −2.28449468547575727739131879572, −1.38311022151609833671519655901, 0.97550993094864773449085357313, 1.95227573563816598773577682010, 3.12553722017391096522353733463, 3.90733929915604425044142528201, 5.12129337792010114511860727811, 5.39074675916601984766142571676, 6.008047801952904466951332486298, 7.40351073595527027167983583554, 8.70008686932320107133005345528, 9.504817547612865920440065071444, 10.37192035503994331870121523744, 10.82526825040077339259755917976, 11.69341493251124055446802826774, 12.40667938363663622718126351066, 13.54114748654079105271201596951, 14.023010096672223868314926772887, 14.83493534136830320954884889703, 15.592540082916582556069663173362, 16.445959301482659442378564129517, 17.25772353495749770658468051373, 18.04823395146067173970608554930, 18.85130920385660604346025129542, 20.03641610528614644932713653098, 20.87029273499862719842509528956, 21.13895912055851984543814440420

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