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Properties

Label 1-127-127.38-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.895 + 0.445i$
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  + (0.623 + 0.781i)2-s + (0.0747 + 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.222 − 0.974i)5-s + (−0.733 + 0.680i)6-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.623 − 0.781i)10-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)12-s + (−0.733 + 0.680i)13-s + (−0.988 − 0.149i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.0747 + 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.222 − 0.974i)5-s + (−0.733 + 0.680i)6-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.623 − 0.781i)10-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)12-s + (−0.733 + 0.680i)13-s + (−0.988 − 0.149i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.955 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2707020852 + 1.152195972i\)
\(L(\frac12)\) \(\approx\) \(0.2707020852 + 1.152195972i\)
\(L(1)\) \(\approx\) \(0.7926291260 + 0.8964133094i\)
\(L(1)\) \(\approx\) \(0.7926291260 + 0.8964133094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.0747 + 0.997i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (0.955 - 0.294i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (0.0747 + 0.997i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.365 - 0.930i)T \)
83 \( 1 + (-0.988 + 0.149i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.826 + 0.563i)T \)
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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

−28.98742985289505278300483585485, −27.45051297744048450817990205063, −26.5091016389645526946732883443, −25.19623851447654012221025843195, −24.16860975957863991786219271937, −23.00799247700475451034863846948, −22.69369669571444040955308436722, −21.44120134140261467111900842339, −19.885868025344212860277903524730, −19.43214327633267326062953112623, −18.57391669784800773928064562745, −17.45400428518621727397521508736, −15.8011374955887952824850873780, −14.261760141692410641858862779966, −13.90800310616534728376174327551, −12.661445190222036435479101880173, −11.709292222411732351210929729293, −10.68230730501847646758304700568, −9.5375848929617584123709198653, −7.67573584466897211107125576743, −6.642863247435803776154481439219, −5.54009168729892797578517620787, −3.43136334220549099371649747125, −2.87021627766451126212074153163, −0.93767898959934993025580146416, 2.85856665499730420661985659595, 4.27068179895533035737693435492, 5.029387705600024087622906616695, 6.2392425601360228397512279277, 7.83968399875760013627512259428, 9.0746776755325716245513935696, 9.75296180806021835485735239865, 11.93612950597581996620237586416, 12.402230502295848325549466514071, 13.92430389618255333500746112442, 14.995014909432727143753390786096, 15.84362879718602240921620230578, 16.61058213211734377538551651921, 17.42657024806781408334586126945, 19.262894721932939643691756085536, 20.51290596472787842002718797044, 21.293320688382046588556638318485, 22.36368893187748766575516818387, 23.020748024426064020943051773559, 24.3818206968208291399007993978, 25.21209049460978067956971698375, 26.09210861591890999274513669580, 27.13510432067517680933841432695, 28.16102685423764275336014277375, 28.99489657007658787725506077907

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