↑

Properties

Label 1-127-127.32-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.870 + 0.493i$
Analytic cond. $0.589785$
Root an. cond. $0.589785$
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  + (−0.222 − 0.974i)2-s + (0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.222 + 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.222 + 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.870 + 0.493i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.870 + 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1385256985 - 0.5253944398i\)
\(L(\frac12)\) \(\approx\) \(-0.1385256985 - 0.5253944398i\)
\(L(1)\) \(\approx\) \(0.4079646230 - 0.5636961795i\)
\(L(1)\) \(\approx\) \(0.4079646230 - 0.5636961795i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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

−29.17042312730489826601740615797, −28.01771253600320771013921968110, −26.9206908005516345216799565338, −26.56251405607951761695448600818, −25.62643147615513411143473422738, −24.52072438027683367517674702850, −23.53674141992633892590412381737, −22.14322990476998848984567219477, −21.91799686952968545374121650134, −19.80777345437576633989496277938, −19.35177691768330889604130538141, −18.26424579250389554051583324019, −16.57323665939062208881223803651, −15.964979651753696893136091422975, −15.17842754432560971311282429234, −14.23835125207886398440415958258, −13.09344967335346608467978907883, −11.35390446504992189901911967866, −10.0042942317137444978697255078, −9.064127892499456206388233718059, −8.0455871643048557567333081617, −6.9217139523123638399955539660, −5.48222611564691115495852383022, −4.10068333703074847374240687305, −2.99875096461253298322070730497, 0.49166113383411452659188966786, 2.32264618485558453640198002586, 3.47657734488511158189499301037, 4.72794754031701962269474899697, 7.03282033662053630355446378848, 7.88749089145798101154940315212, 9.12243051971205271293512605379, 10.077261794402081820155560930983, 11.68905346184688660959675242657, 12.563352371063655503964111943836, 13.17043990616323495357583695693, 14.4878779294811290906128672712, 15.86402740539499837120229242389, 17.29170930304007788556519376482, 18.3192466121240198086527596303, 19.38066819615351761736450649371, 20.10772076023708093188575838292, 20.46615323236376151447590020563, 22.3516969531018677401256528278, 23.04349774010693021920723237093, 24.15496238626629243324064205279, 25.36894751793250682086459750702, 26.51352296085736155620175251979, 27.13950500298477659249547871136, 28.68459995558628948434924907147

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