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Properties

Label 1-127-127.124-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.930 + 0.365i$
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 + (−0.998 − 0.0498i)3-s + (0.623 + 0.781i)4-s + (0.365 + 0.930i)5-s + (0.878 + 0.478i)6-s + (−0.853 + 0.521i)7-s + (−0.222 − 0.974i)8-s + (0.995 + 0.0995i)9-s + (0.0747 − 0.997i)10-s + (−0.969 − 0.246i)11-s + (−0.583 − 0.811i)12-s + (−0.0249 − 0.999i)13-s + (0.995 − 0.0995i)14-s + (−0.318 − 0.947i)15-s + (−0.222 + 0.974i)16-s + (−0.318 + 0.947i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.998 − 0.0498i)3-s + (0.623 + 0.781i)4-s + (0.365 + 0.930i)5-s + (0.878 + 0.478i)6-s + (−0.853 + 0.521i)7-s + (−0.222 − 0.974i)8-s + (0.995 + 0.0995i)9-s + (0.0747 − 0.997i)10-s + (−0.969 − 0.246i)11-s + (−0.583 − 0.811i)12-s + (−0.0249 − 0.999i)13-s + (0.995 − 0.0995i)14-s + (−0.318 − 0.947i)15-s + (−0.222 + 0.974i)16-s + (−0.318 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.930 + 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02017367659 + 0.1064784030i\)
\(L(\frac12)\) \(\approx\) \(0.02017367659 + 0.1064784030i\)
\(L(1)\) \(\approx\) \(0.3690024558 + 0.03420026198i\)
\(L(1)\) \(\approx\) \(0.3690024558 + 0.03420026198i\)

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 \( 1 + (-0.998 - 0.0498i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.853 + 0.521i)T \)
11 \( 1 + (-0.969 - 0.246i)T \)
13 \( 1 + (-0.0249 - 0.999i)T \)
17 \( 1 + (-0.318 + 0.947i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.969 + 0.246i)T \)
29 \( 1 + (-0.797 + 0.603i)T \)
31 \( 1 + (-0.661 + 0.749i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.661 - 0.749i)T \)
43 \( 1 + (0.921 + 0.388i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (-0.583 + 0.811i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (0.542 - 0.840i)T \)
71 \( 1 + (0.698 + 0.715i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.270 - 0.962i)T \)
83 \( 1 + (-0.411 - 0.911i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 + (-0.124 + 0.992i)T \)
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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

−28.49245673044596290657779958997, −27.505782352300274775056000409064, −26.429399464436221584539090401309, −25.516461741353377060959748885531, −24.2541924798697981357776075821, −23.66624859699543883320277466617, −22.59258414329872890590131666288, −21.08750922572490823560922276675, −20.24873000062928791358484800268, −18.8712425489475338737494705725, −18.04509784466146129164172801473, −16.76398556640327544404791452009, −16.50586218222972256087184422893, −15.525220881717580170252924853, −13.7069472972443857419153298278, −12.5512004585105680332400947410, −11.34532067913305345451829902318, −10.042011157266424980219449536856, −9.49226905050909385973453709652, −7.89702880505619888549840470902, −6.66664344299031652740656081412, −5.708698489827385596729429384823, −4.47179478345819193933445645005, −1.86364402638065903315164496523, −0.13936738676045895098173660963, 2.151970055927858480012563963708, 3.43463465531772518629775481039, 5.66861629198549492807914972097, 6.591029834883374771789695395300, 7.75961787556557790394892537530, 9.37871746772906309328525415520, 10.54942080590598247028165780678, 10.89587592721354795177991385962, 12.44083165343044033887652151358, 13.128652224076667406775853971151, 15.31423662595123856465651780540, 16.0026506429591814852176381619, 17.32588864500058631042896979589, 18.083653798445994383851821141595, 18.78925352730991813526079047181, 19.851586994983545658350677397133, 21.53386523248634133001959936857, 21.93048601590501083933539053672, 23.01566396207181634659953203110, 24.34895351765018717387373767907, 25.720224738500189887586974340960, 26.21928694322574289115560471153, 27.489910207738687034372526520551, 28.35127949783988053617091826204, 29.16787696409811848044269345812

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