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Properties

Label 1-633-633.173-r1-0-0
Degree $1$
Conductor $633$
Sign $0.847 - 0.531i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.900 − 0.433i)5-s + (0.826 + 0.563i)7-s + (0.222 + 0.974i)8-s + (−0.5 − 0.866i)10-s + (0.900 + 0.433i)11-s + (−0.222 + 0.974i)13-s + (0.5 − 0.866i)14-s + (0.955 − 0.294i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)20-s + (0.365 − 0.930i)22-s − 23-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.900 − 0.433i)5-s + (0.826 + 0.563i)7-s + (0.222 + 0.974i)8-s + (−0.5 − 0.866i)10-s + (0.900 + 0.433i)11-s + (−0.222 + 0.974i)13-s + (0.5 − 0.866i)14-s + (0.955 − 0.294i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)20-s + (0.365 − 0.930i)22-s − 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.847 - 0.531i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ 0.847 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.465230807 - 0.7088419071i\)
\(L(\frac12)\) \(\approx\) \(2.465230807 - 0.7088419071i\)
\(L(1)\) \(\approx\) \(1.232691439 - 0.4602805601i\)
\(L(1)\) \(\approx\) \(1.232691439 - 0.4602805601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (-0.733 + 0.680i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (0.955 + 0.294i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (0.733 - 0.680i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−22.917111956051278471122054795469, −22.0614584328613291214334778482, −21.44687311524686786172543761398, −20.39962274182341512036969668637, −19.29518080640323852892351677769, −18.401717912355681671761109340110, −17.62333506593188693244126388989, −17.08090646210622211430115172398, −16.4059525274216353455078288983, −15.0149283813106320575263697398, −14.552899319060482110263919037421, −13.89181360068586502512593166480, −13.05871777771173624948652604072, −11.93035606727102984215628960984, −10.49681995081228617186985821084, −10.119298346544650728769652125466, −8.94911890988381512353900569893, −8.02198509828810389628623002377, −7.3248160694610581668226109304, −6.01257627431810471173168861002, −5.79270175008131342699707935642, −4.42153997348480175939739797925, −3.5280574401462396721348987089, −1.86163574274605843917981936711, −0.74531049588397824903030053875, 1.06637774749276294572724234544, 1.86320642675730157177801091120, 2.68118203477949661045844255590, 4.21397872984921099487343339130, 4.8787219553598764300913166101, 5.82848844285326346524405178930, 7.13061778354286639896696517255, 8.528969203858858323158289197043, 9.096208908434717104117451609017, 9.80975778589251111543136199998, 10.81208962439384578586780727095, 11.93177687305614956948001035101, 12.17961769221396340145617779736, 13.40793122760620532474046409817, 14.21069214425830794317055275162, 14.69546546729910924111329113674, 16.2851059646479493652205419028, 17.16841682942317995679939825262, 17.83408628253635320474252057681, 18.48482784587616070626601973495, 19.50139642705746018991887551711, 20.32030233673859428998567178664, 21.02839360992079208955719280784, 21.86801107429945165962733141629, 22.07525732981501356659056850154

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