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Properties

Label 1-177-177.26-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.249 - 0.968i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (−0.267 + 0.963i)2-s + (−0.856 − 0.515i)4-s + (−0.0541 + 0.998i)5-s + (−0.161 + 0.986i)7-s + (0.725 − 0.687i)8-s + (−0.947 − 0.319i)10-s + (−0.468 + 0.883i)11-s + (−0.370 − 0.928i)13-s + (−0.907 − 0.419i)14-s + (0.468 + 0.883i)16-s + (0.161 + 0.986i)17-s + (0.647 + 0.762i)19-s + (0.561 − 0.827i)20-s + (−0.725 − 0.687i)22-s + (−0.796 + 0.605i)23-s + ⋯
L(s)  = 1  + (−0.267 + 0.963i)2-s + (−0.856 − 0.515i)4-s + (−0.0541 + 0.998i)5-s + (−0.161 + 0.986i)7-s + (0.725 − 0.687i)8-s + (−0.947 − 0.319i)10-s + (−0.468 + 0.883i)11-s + (−0.370 − 0.928i)13-s + (−0.907 − 0.419i)14-s + (0.468 + 0.883i)16-s + (0.161 + 0.986i)17-s + (0.647 + 0.762i)19-s + (0.561 − 0.827i)20-s + (−0.725 − 0.687i)22-s + (−0.796 + 0.605i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.249 - 0.968i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.249 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3131465763 + 0.4041211045i\)
\(L(\frac12)\) \(\approx\) \(-0.3131465763 + 0.4041211045i\)
\(L(1)\) \(\approx\) \(0.4677311196 + 0.5102151445i\)
\(L(1)\) \(\approx\) \(0.4677311196 + 0.5102151445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.267 + 0.963i)T \)
5 \( 1 + (-0.0541 + 0.998i)T \)
7 \( 1 + (-0.161 + 0.986i)T \)
11 \( 1 + (-0.468 + 0.883i)T \)
13 \( 1 + (-0.370 - 0.928i)T \)
17 \( 1 + (0.161 + 0.986i)T \)
19 \( 1 + (0.647 + 0.762i)T \)
23 \( 1 + (-0.796 + 0.605i)T \)
29 \( 1 + (-0.267 - 0.963i)T \)
31 \( 1 + (0.647 - 0.762i)T \)
37 \( 1 + (-0.725 - 0.687i)T \)
41 \( 1 + (-0.796 - 0.605i)T \)
43 \( 1 + (0.468 + 0.883i)T \)
47 \( 1 + (-0.0541 - 0.998i)T \)
53 \( 1 + (0.947 - 0.319i)T \)
61 \( 1 + (0.267 - 0.963i)T \)
67 \( 1 + (-0.725 + 0.687i)T \)
71 \( 1 + (-0.0541 - 0.998i)T \)
73 \( 1 + (0.907 + 0.419i)T \)
79 \( 1 + (-0.561 + 0.827i)T \)
83 \( 1 + (-0.976 - 0.214i)T \)
89 \( 1 + (-0.267 - 0.963i)T \)
97 \( 1 + (0.907 - 0.419i)T \)
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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

−26.74055171234306412525100287429, −25.86215742273983225328656432247, −24.292797628264773117920995584, −23.65066663828239854113712461369, −22.43212532930763818263421722229, −21.40427431584522740929309430964, −20.52636060967572246548466364296, −19.87503270504883455564272546920, −18.89141595126633291717261541920, −17.75820333216236185044489819145, −16.67743835023380537713893662926, −16.12412618728207571406278394267, −13.955287575176291992587557086465, −13.557508138481781115712136688480, −12.32189271486684821860815157526, −11.45092398263028722747103022240, −10.30425439957293367470543424268, −9.295560252700627295348305654410, −8.36060904482360218475242234937, −7.14306256278301149047692042059, −5.17560137354292693519440714492, −4.26923984697523897249021305886, −2.97188034289378401257761579773, −1.30211077337974652943113033463, −0.2140045526408624702155331154, 2.143258849654070316903367308259, 3.720812148891023067752088503850, 5.38722404181610210145683405653, 6.14778433017050952963272307167, 7.45233738937374889258962852709, 8.18748846299386444307447841140, 9.735321069872878995126192722349, 10.30711328430359760546815860139, 11.95347851444960265406955998737, 13.10248447834580672538131966156, 14.38876599288774370032348234569, 15.24234564203030912123298688953, 15.688414028557752668920800156258, 17.25580384472290966635007351276, 18.04745273278198949534594411959, 18.77497746388501237905648397727, 19.74793417988915897183499394895, 21.364356154323957771460433633805, 22.514312808367514408671608929183, 22.86695395215268667313731801625, 24.17317323262610206862074149929, 25.15753109622240467634309455256, 25.85098964855430113281505694885, 26.63086480935173042883160822515, 27.76757093002470800581948064856

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