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Properties

Label 1-127-127.36-r0-0-0
Degree $1$
Conductor $127$
Sign $0.506 + 0.862i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.542 + 0.840i)3-s + (0.623 − 0.781i)4-s + (0.365 − 0.930i)5-s + (−0.853 − 0.521i)6-s + (−0.0249 + 0.999i)7-s + (−0.222 + 0.974i)8-s + (−0.411 + 0.911i)9-s + (0.0747 + 0.997i)10-s + (0.270 − 0.962i)11-s + (0.995 + 0.0995i)12-s + (0.878 − 0.478i)13-s + (−0.411 − 0.911i)14-s + (0.980 − 0.198i)15-s + (−0.222 − 0.974i)16-s + (0.980 + 0.198i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.542 + 0.840i)3-s + (0.623 − 0.781i)4-s + (0.365 − 0.930i)5-s + (−0.853 − 0.521i)6-s + (−0.0249 + 0.999i)7-s + (−0.222 + 0.974i)8-s + (−0.411 + 0.911i)9-s + (0.0747 + 0.997i)10-s + (0.270 − 0.962i)11-s + (0.995 + 0.0995i)12-s + (0.878 − 0.478i)13-s + (−0.411 − 0.911i)14-s + (0.980 − 0.198i)15-s + (−0.222 − 0.974i)16-s + (0.980 + 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8005853030 + 0.4580751952i\)
\(L(\frac12)\) \(\approx\) \(0.8005853030 + 0.4580751952i\)
\(L(1)\) \(\approx\) \(0.8511114844 + 0.3228922194i\)
\(L(1)\) \(\approx\) \(0.8511114844 + 0.3228922194i\)

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 \( 1 + (0.542 + 0.840i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
7 \( 1 + (-0.0249 + 0.999i)T \)
11 \( 1 + (0.270 - 0.962i)T \)
13 \( 1 + (0.878 - 0.478i)T \)
17 \( 1 + (0.980 + 0.198i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.270 + 0.962i)T \)
29 \( 1 + (0.921 - 0.388i)T \)
31 \( 1 + (-0.318 + 0.947i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.318 - 0.947i)T \)
43 \( 1 + (-0.124 + 0.992i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.995 - 0.0995i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.456 - 0.889i)T \)
71 \( 1 + (-0.969 - 0.246i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (0.698 - 0.715i)T \)
83 \( 1 + (-0.583 - 0.811i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (-0.797 + 0.603i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.85745361482898297302264408548, −27.624947136824035640634989284255, −26.34612920508587485089434358322, −25.91104402262917624095766866073, −25.118495974830818724057207856742, −23.684701682603665256177641781522, −22.698678612074731274724720071910, −21.163928042548629351817024058542, −20.328187917126177535215778763280, −19.36321247346400360313156789174, −18.49664045785996359203427488505, −17.71056307563460720136019901098, −16.75586999719961838335065274989, −15.1033212922131896306799271583, −13.98819196224937877133069458904, −12.96665341658988344360117970113, −11.697697713328019619513754550274, −10.562233913508175022565010441313, −9.57622088428778248528442276258, −8.282515206991683735447841860442, −7.052591942792520749007637982, −6.63286568552807656181171164596, −3.813381514971687351335153774142, −2.58721764913839455647667163965, −1.32826408067061228408658372927, 1.61497494091747395694295035274, 3.31711032212093096079776554980, 5.279446342473404398897959815106, 5.965371905099960115460840774804, 8.19467880798934598055390005798, 8.634391365920813942965238438166, 9.59766068826721361620269195592, 10.66888085421253340383055482968, 12.02835275525279930479767493326, 13.67613725445090852557175639862, 14.79156321924650246984278145541, 15.9190327761685815580044884868, 16.412464077947178005333988015933, 17.58892609786874392940593410872, 18.90613800707805988715226165263, 19.70332088624295764079756545831, 21.04049762672808429755484437590, 21.36029857581350044343965755171, 23.110606433457564746317060442555, 24.48963845956565350223721475015, 25.27226241921974052297915533177, 25.77064033078005631904162086825, 27.27797692755426613527439574043, 27.71194337776556813695837322881, 28.580493208704585472977813433

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