↑

Properties

Label 1-107-107.63-r1-0-0
Degree $1$
Conductor $107$
Sign $0.619 - 0.785i$
Analytic cond. $11.4987$
Root an. cond. $11.4987$
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  + (−0.147 − 0.989i)2-s + (0.582 − 0.812i)3-s + (−0.956 + 0.292i)4-s + (0.630 + 0.776i)5-s + (−0.889 − 0.456i)6-s + (0.0887 + 0.996i)7-s + (0.430 + 0.902i)8-s + (−0.320 − 0.947i)9-s + (0.674 − 0.737i)10-s + (0.992 − 0.118i)11-s + (−0.320 + 0.947i)12-s + (0.482 + 0.875i)13-s + (0.972 − 0.234i)14-s + (0.998 − 0.0592i)15-s + (0.829 − 0.558i)16-s + (0.915 + 0.403i)17-s + ⋯
L(s)  = 1  + (−0.147 − 0.989i)2-s + (0.582 − 0.812i)3-s + (−0.956 + 0.292i)4-s + (0.630 + 0.776i)5-s + (−0.889 − 0.456i)6-s + (0.0887 + 0.996i)7-s + (0.430 + 0.902i)8-s + (−0.320 − 0.947i)9-s + (0.674 − 0.737i)10-s + (0.992 − 0.118i)11-s + (−0.320 + 0.947i)12-s + (0.482 + 0.875i)13-s + (0.972 − 0.234i)14-s + (0.998 − 0.0592i)15-s + (0.829 − 0.558i)16-s + (0.915 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.619 - 0.785i$
Analytic conductor: \(11.4987\)
Root analytic conductor: \(11.4987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (1:\ ),\ 0.619 - 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.892710494 - 0.9179890727i\)
\(L(\frac12)\) \(\approx\) \(1.892710494 - 0.9179890727i\)
\(L(1)\) \(\approx\) \(1.229644821 - 0.5573185743i\)
\(L(1)\) \(\approx\) \(1.229644821 - 0.5573185743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.147 - 0.989i)T \)
3 \( 1 + (0.582 - 0.812i)T \)
5 \( 1 + (0.630 + 0.776i)T \)
7 \( 1 + (0.0887 + 0.996i)T \)
11 \( 1 + (0.992 - 0.118i)T \)
13 \( 1 + (0.482 + 0.875i)T \)
17 \( 1 + (0.915 + 0.403i)T \)
19 \( 1 + (-0.533 - 0.845i)T \)
23 \( 1 + (0.972 + 0.234i)T \)
29 \( 1 + (0.0296 + 0.999i)T \)
31 \( 1 + (-0.757 - 0.652i)T \)
37 \( 1 + (-0.794 - 0.606i)T \)
41 \( 1 + (0.937 + 0.348i)T \)
43 \( 1 + (0.630 - 0.776i)T \)
47 \( 1 + (0.937 - 0.348i)T \)
53 \( 1 + (0.147 - 0.989i)T \)
59 \( 1 + (-0.0296 + 0.999i)T \)
61 \( 1 + (-0.0887 + 0.996i)T \)
67 \( 1 + (0.430 - 0.902i)T \)
71 \( 1 + (-0.582 - 0.812i)T \)
73 \( 1 + (-0.263 + 0.964i)T \)
79 \( 1 + (0.375 + 0.926i)T \)
83 \( 1 + (-0.984 - 0.176i)T \)
89 \( 1 + (0.889 - 0.456i)T \)
97 \( 1 + (-0.674 + 0.737i)T \)
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

−29.5030882465812533020297743642, −27.85692734090623995299139742964, −27.46321016573613237753181054410, −26.35188472314310491967898047068, −25.22748854947655418576350729329, −24.90389219183345467199026663752, −23.35014241326349967264310788168, −22.46981824337951429950633437226, −21.11081198249372959337881852102, −20.30060538231243015977974240143, −19.08474277342891651720259645866, −17.39193995954773079832180646998, −16.830366499351127939394402881350, −15.90362310587402121022103571502, −14.55805699228434235593047745605, −13.91284129270024102574276569006, −12.79045169705540725549005685366, −10.552770352685359595731674169426, −9.643141243333233460469864183904, −8.67286978900501473351382498778, −7.59870517942984410426590572731, −5.95750344604496563931723858385, −4.75444879262398041416502993116, −3.69106458113494359281947473466, −1.10402272030372012930397444343, 1.44973376404355288889639465370, 2.46489987724192493920300336098, 3.64298059306620021767975157791, 5.76199922132304527526255907862, 7.08764341294167277521691943773, 8.80504660155236797439502387030, 9.29966428733170018702315601570, 10.97252507843844097920224803744, 11.97444931336714012372479997943, 13.035370617217611609369632603, 14.15230505149985691462046232874, 14.835576235904031958832144981449, 17.07990158745631836182015560550, 18.11116358438976882006583765170, 18.90925940536730593199116988415, 19.49430033652281480190275735330, 21.02843531589819234929020378824, 21.68711232831470399371063283722, 22.80377311312795018866886565495, 24.07376532451660379536445818371, 25.48793210392134232695261590285, 25.94012794538215615014650802334, 27.3150306770767801931392045873, 28.44484961466278814967057251428, 29.42778497773259835993135257482

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