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Properties

Label 1-127-127.107-r0-0-0
Degree $1$
Conductor $127$
Sign $0.631 + 0.775i$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.631 + 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599123569 + 0.7601345913i\)
\(L(\frac12)\) \(\approx\) \(1.599123569 + 0.7601345913i\)
\(L(1)\) \(\approx\) \(1.593051793 + 0.4825625733i\)
\(L(1)\) \(\approx\) \(1.593051793 + 0.4825625733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.97954151234807838730548298599, −28.30131467808814782224997790430, −26.22928034791490873331438198739, −25.43155533259473546171050581203, −24.47134516064402092511024226569, −23.65488271149275689463566631541, −22.53445288973440017085418500036, −22.11562446449883195879909399907, −20.47305556030356031880959391624, −19.92040194916778137921220772375, −18.32811443124747839044628449894, −17.29362339811755813117252385896, −16.484229034263089311298565902147, −14.960974856794992779550797277594, −13.774235207414055700775747702246, −13.033127168405569893775416697614, −12.38950678609200081255276090971, −10.811868335993979379087391818499, −10.03168458489605115173147139481, −7.74679567085763228870234775866, −6.80389362138512352427257857336, −5.82580166666513092899911508508, −4.74290463411232324617307409082, −2.87399975823113184516918401259, −1.60693526792258478686238558541, 2.31191146818875603792717173486, 3.52174276344467370190395728180, 5.19489595165270410555058805439, 5.66320414680177169003296094872, 6.86318379600857400100381986650, 9.0536282325265148658189617177, 9.9965031368050814779787660210, 11.27127926485812903117166208441, 12.14083481676412797522110368855, 13.50481735397668032200172993720, 14.34022646714235918763240685250, 15.709393975371444824772844777228, 16.21602787706684006472815000707, 17.44033802958721446486018313457, 18.800714698456973016974206097324, 20.3263343566691609464485175463, 21.34117152006993513804413975710, 21.90146984861078464059627098464, 22.5264919069838600658651460976, 23.88036417335994357769061310847, 24.83372091770627525974147903663, 25.87255940652954545182529565084, 26.818334274544364137776579196441, 28.56291430046158493044098691792, 28.87821198029464717937106505755

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