↑

Properties

Label 1-127-127.72-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.996 + 0.0876i$
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  + (−0.900 − 0.433i)2-s + (−0.124 − 0.992i)3-s + (0.623 + 0.781i)4-s + (−0.988 − 0.149i)5-s + (−0.318 + 0.947i)6-s + (0.980 − 0.198i)7-s + (−0.222 − 0.974i)8-s + (−0.969 + 0.246i)9-s + (0.826 + 0.563i)10-s + (−0.583 − 0.811i)11-s + (0.698 − 0.715i)12-s + (−0.661 − 0.749i)13-s + (−0.969 − 0.246i)14-s + (−0.0249 + 0.999i)15-s + (−0.222 + 0.974i)16-s + (−0.0249 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.124 − 0.992i)3-s + (0.623 + 0.781i)4-s + (−0.988 − 0.149i)5-s + (−0.318 + 0.947i)6-s + (0.980 − 0.198i)7-s + (−0.222 − 0.974i)8-s + (−0.969 + 0.246i)9-s + (0.826 + 0.563i)10-s + (−0.583 − 0.811i)11-s + (0.698 − 0.715i)12-s + (−0.661 − 0.749i)13-s + (−0.969 − 0.246i)14-s + (−0.0249 + 0.999i)15-s + (−0.222 + 0.974i)16-s + (−0.0249 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.996 + 0.0876i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.996 + 0.0876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01589492324 - 0.3620247784i\)
\(L(\frac12)\) \(\approx\) \(0.01589492324 - 0.3620247784i\)
\(L(1)\) \(\approx\) \(0.3825785771 - 0.3302393620i\)
\(L(1)\) \(\approx\) \(0.3825785771 - 0.3302393620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (-0.124 - 0.992i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.980 - 0.198i)T \)
11 \( 1 + (-0.583 - 0.811i)T \)
13 \( 1 + (-0.661 - 0.749i)T \)
17 \( 1 + (-0.0249 - 0.999i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.583 + 0.811i)T \)
29 \( 1 + (-0.998 - 0.0498i)T \)
31 \( 1 + (-0.853 - 0.521i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.853 + 0.521i)T \)
43 \( 1 + (0.542 - 0.840i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (0.698 + 0.715i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (-0.797 + 0.603i)T \)
71 \( 1 + (-0.411 - 0.911i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.995 + 0.0995i)T \)
83 \( 1 + (0.270 - 0.962i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (0.456 - 0.889i)T \)
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

−28.743543282254811746310124408666, −28.00732626302440833348384343711, −27.39996877343309923469071672773, −26.41607504581901007400545687819, −25.84631826148332021434730005866, −24.15201855505261074311122955593, −23.67073790254077813967640736131, −22.30633701657678729835212737143, −21.01906204378557910083898347975, −20.151219992713492169550335788724, −19.16746267780721315002313915026, −17.91231060397497593980736198007, −17.00164537468789179454398712823, −15.95921637862654215995413862714, −14.99393881475190390057804806473, −14.601675658897474182940626188659, −12.13461599251876412421272350790, −11.11305910940011946095611612351, −10.36012884263131603104050001811, −8.99360190051631401455883969522, −8.11352201591683935388398949476, −6.94646693519797590129302178611, −5.244530410696022168392030673327, −4.25154989824130090769781437091, −2.24648226862280451609322778411, 0.43222652817889067800592148255, 2.02451720389035086784156247203, 3.50118256435611122683953155098, 5.44696106823639282391525651100, 7.34035017541417000521375355860, 7.81115303755905513044889196394, 8.73341330138561614251932481804, 10.57672647715825798206640008904, 11.5053978190138553145717111047, 12.20809938798623312311098324435, 13.42649957724702654227894402049, 14.936656304565633894718569000570, 16.286739437262160472839924730551, 17.260245455213713622381532256875, 18.308800385203583665440470293540, 18.966534769442160883876745665322, 20.04596254285110461978134088905, 20.7353276593989368974180949981, 22.263526690769928342063771457926, 23.60309634342191257972331688109, 24.35540235977572346564788131638, 25.24017429499703724204821256589, 26.62617346617062954006139314690, 27.39973872087543459627186774223, 28.17112929358362910818697865696

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