↑

Properties

Label 1-127-127.73-r0-0-0
Degree $1$
Conductor $127$
Sign $0.916 - 0.400i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (0.826 + 0.563i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (0.955 − 0.294i)6-s + (0.955 − 0.294i)7-s + (−0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.623 + 0.781i)10-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)12-s + (0.955 − 0.294i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)15-s + (−0.900 + 0.433i)16-s + (−0.733 − 0.680i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (0.826 + 0.563i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (0.955 − 0.294i)6-s + (0.955 − 0.294i)7-s + (−0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.623 + 0.781i)10-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)12-s + (0.955 − 0.294i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)15-s + (−0.900 + 0.433i)16-s + (−0.733 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.754179087 - 0.3667165053i\)
\(L(\frac12)\) \(\approx\) \(1.754179087 - 0.3667165053i\)
\(L(1)\) \(\approx\) \(1.648242174 - 0.3077156113i\)
\(L(1)\) \(\approx\) \(1.648242174 - 0.3077156113i\)

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

Imaginary part of the first few zeros on the critical line

−28.99754762722097812175114493421, −27.75670807756344372911810874384, −26.50425794059153147021429016811, −25.738760817213458543168862519286, −24.5711791535825526503314754210, −24.09114562690042704586831451835, −23.42076106869138120904908052892, −21.65738174772966663469795894991, −20.83724530239383236920648757245, −20.087713126480057777895442141453, −18.38333983479068684045148274331, −17.72843397040008576525902262049, −16.19345817670614354272351026021, −15.452160284254958859410363220328, −14.28039986729641030002517229376, −13.364862810513324895229285386853, −12.57803223848734692440352674305, −11.404250067982474379307801024075, −9.11743502934874425511276034776, −8.26768129213346429835960872668, −7.638413960896795371645817267121, −6.008250659971846957406929867720, −4.79994440740336179200254892533, −3.55686927903822097668778053942, −1.8328430834666030734161505253, 2.0327355568841874801224457750, 3.14224166441449602320003957221, 4.21835778505013254517680595550, 5.44472170076379921178555231092, 7.30136656430100095157590088459, 8.55121588696500492837690952785, 10.10210723767548345972204651757, 10.741006635430735912559705738415, 11.7372931348655859453539902333, 13.59939808929391648465391271225, 13.93692396608011029066368652069, 15.204329873871726676486490648523, 15.7536733349424204428964970121, 18.070169956755765840834795814880, 18.61932466766481434436397870398, 20.071802094723942712634146816452, 20.587228820950011981037609078240, 21.58041081673757882243916459767, 22.506349429955179513299426349121, 23.52587040580810318597256293601, 24.620595103808567352529877379372, 26.00725858421515254651650030739, 26.89055447723070323016780191312, 27.658167892233304505734873621021, 28.85434564689309967189540569511

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