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Properties

Label 1-127-127.49-r0-0-0
Degree $1$
Conductor $127$
Sign $0.336 - 0.941i$
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 + (0.456 − 0.889i)3-s + (0.623 − 0.781i)4-s + (0.365 − 0.930i)5-s + (−0.0249 + 0.999i)6-s + (0.878 − 0.478i)7-s + (−0.222 + 0.974i)8-s + (−0.583 − 0.811i)9-s + (0.0747 + 0.997i)10-s + (0.698 + 0.715i)11-s + (−0.411 − 0.911i)12-s + (−0.853 − 0.521i)13-s + (−0.583 + 0.811i)14-s + (−0.661 − 0.749i)15-s + (−0.222 − 0.974i)16-s + (−0.661 + 0.749i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.456 − 0.889i)3-s + (0.623 − 0.781i)4-s + (0.365 − 0.930i)5-s + (−0.0249 + 0.999i)6-s + (0.878 − 0.478i)7-s + (−0.222 + 0.974i)8-s + (−0.583 − 0.811i)9-s + (0.0747 + 0.997i)10-s + (0.698 + 0.715i)11-s + (−0.411 − 0.911i)12-s + (−0.853 − 0.521i)13-s + (−0.583 + 0.811i)14-s + (−0.661 − 0.749i)15-s + (−0.222 − 0.974i)16-s + (−0.661 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7414368327 - 0.5225561340i\)
\(L(\frac12)\) \(\approx\) \(0.7414368327 - 0.5225561340i\)
\(L(1)\) \(\approx\) \(0.8493013570 - 0.2983528097i\)
\(L(1)\) \(\approx\) \(0.8493013570 - 0.2983528097i\)

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 \( 1 + (0.456 - 0.889i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
7 \( 1 + (0.878 - 0.478i)T \)
11 \( 1 + (0.698 + 0.715i)T \)
13 \( 1 + (-0.853 - 0.521i)T \)
17 \( 1 + (-0.661 + 0.749i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.698 - 0.715i)T \)
29 \( 1 + (-0.124 + 0.992i)T \)
31 \( 1 + (0.980 - 0.198i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.980 + 0.198i)T \)
43 \( 1 + (-0.797 - 0.603i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.411 + 0.911i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.998 + 0.0498i)T \)
71 \( 1 + (0.270 + 0.962i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (-0.969 - 0.246i)T \)
83 \( 1 + (0.995 - 0.0995i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.921 + 0.388i)T \)
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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

−28.89572627827230563233754532854, −27.74894665590735800609324119924, −26.933572435776900646258378569242, −26.461853744097825993410070104429, −25.26665134967325218167804410491, −24.51047064928293291195015953006, −22.429738416909950411615053084796, −21.60233743071829745382446129920, −21.11981217931941200579654807667, −19.691827893143529735315264473684, −19.0205378346329245968201910144, −17.737966857027793555429440192470, −16.956486141606418858085893169362, −15.578435671608309792021661514395, −14.72782358543611260323420475321, −13.60541624027093992009094426699, −11.46913329745808924493512230809, −11.19267908079767404058761460408, −9.76732572101283560503435205374, −9.06030469430817432092072162115, −7.87284190812233974476548643290, −6.49765313725543652768709131153, −4.64998659935291474639560032270, −3.07274760312956859797280431097, −2.13843200485992758819051096103, 1.181018319393170535928524973967, 2.1394805751930097590730440439, 4.62609877820550648863246050773, 6.11426402700667507189896852401, 7.32640210254192347682388344273, 8.26247635935819573007763864693, 9.108898073809128113430440387623, 10.394163718125829628563108527277, 11.90337633734712642358282968287, 12.92057548612960374241605009591, 14.36675999364645316826416992064, 14.993474096903798261337907815963, 16.81851568265926220097161468268, 17.36011526337984238240972680233, 18.17112596267534599105563987238, 19.602453032021740637733654603307, 20.124012516302232333418628714224, 21.06658555339160023354960790752, 23.06566015114839298098385659317, 24.1388052972890732347812440592, 24.72326232721301503979740852976, 25.37938575294484662852367610085, 26.61619637848976677224995371689, 27.58755436837202288993057679871, 28.55646956276559809845225466301

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