↑

Properties

Label 1-127-127.30-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: $\chi_{127} (30, \cdot )$
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

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

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