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Properties

Label 1-179-179.139-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.979 - 0.202i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.261 + 0.965i)2-s + (−0.949 − 0.312i)3-s + (−0.863 − 0.505i)4-s + (−0.394 + 0.918i)5-s + (0.550 − 0.835i)6-s + (0.960 − 0.278i)7-s + (0.713 − 0.700i)8-s + (0.804 + 0.593i)9-s + (−0.783 − 0.621i)10-s + (−0.737 + 0.675i)11-s + (0.662 + 0.749i)12-s + (−0.329 + 0.944i)13-s + (0.0176 + 0.999i)14-s + (0.662 − 0.749i)15-s + (0.489 + 0.871i)16-s + (−0.999 + 0.0352i)17-s + ⋯
L(s)  = 1  + (−0.261 + 0.965i)2-s + (−0.949 − 0.312i)3-s + (−0.863 − 0.505i)4-s + (−0.394 + 0.918i)5-s + (0.550 − 0.835i)6-s + (0.960 − 0.278i)7-s + (0.713 − 0.700i)8-s + (0.804 + 0.593i)9-s + (−0.783 − 0.621i)10-s + (−0.737 + 0.675i)11-s + (0.662 + 0.749i)12-s + (−0.329 + 0.944i)13-s + (0.0176 + 0.999i)14-s + (0.662 − 0.749i)15-s + (0.489 + 0.871i)16-s + (−0.999 + 0.0352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.979 - 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03411387282 + 0.3335930237i\)
\(L(\frac12)\) \(\approx\) \(-0.03411387282 + 0.3335930237i\)
\(L(1)\) \(\approx\) \(0.4125075819 + 0.3162370306i\)
\(L(1)\) \(\approx\) \(0.4125075819 + 0.3162370306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.261 + 0.965i)T \)
3 \( 1 + (-0.949 - 0.312i)T \)
5 \( 1 + (-0.394 + 0.918i)T \)
7 \( 1 + (0.960 - 0.278i)T \)
11 \( 1 + (-0.737 + 0.675i)T \)
13 \( 1 + (-0.329 + 0.944i)T \)
17 \( 1 + (-0.999 + 0.0352i)T \)
19 \( 1 + (0.158 - 0.987i)T \)
23 \( 1 + (-0.925 + 0.378i)T \)
29 \( 1 + (-0.984 + 0.175i)T \)
31 \( 1 + (0.760 + 0.648i)T \)
37 \( 1 + (-0.984 - 0.175i)T \)
41 \( 1 + (-0.192 + 0.981i)T \)
43 \( 1 + (-0.688 + 0.725i)T \)
47 \( 1 + (-0.994 - 0.105i)T \)
53 \( 1 + (-0.579 + 0.815i)T \)
59 \( 1 + (-0.0529 - 0.998i)T \)
61 \( 1 + (0.977 + 0.210i)T \)
67 \( 1 + (0.713 + 0.700i)T \)
71 \( 1 + (0.0881 + 0.996i)T \)
73 \( 1 + (-0.635 - 0.772i)T \)
79 \( 1 + (0.227 + 0.973i)T \)
83 \( 1 + (0.997 + 0.0705i)T \)
89 \( 1 + (-0.261 - 0.965i)T \)
97 \( 1 + (-0.896 - 0.442i)T \)
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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

−27.307247481018647015589879457413, −26.41851674461349482657123645534, −24.56948534260029946983323975802, −23.93112991527800281998035753802, −22.759718765492176900166354948824, −21.939314854999124533052879448542, −20.815087430924174282386277167854, −20.49433911763316020437686344639, −19.00137892531247237191362192042, −18.01853963232957511072469421640, −17.29381494019790044268173171412, −16.29112206454930443331798403390, −15.210281875474056915669304771152, −13.54121155324654189294042096687, −12.5130729494871405808354050163, −11.76194901031162161189590418618, −10.89971730780118514743805091951, −9.94607928253214139128408982457, −8.57809287155932686171642570676, −7.81911440980430692816580636894, −5.5891815167507333657962855002, −4.86947399242300348312727963687, −3.7577935647442824109692255113, −1.86897789439473509638434495903, −0.33029038527306885503511448519, 1.87917375136908465328004300550, 4.32410958964314759630414349351, 5.10378463686480045483013782298, 6.58634166435913241385308516406, 7.192327371777912437363595948585, 8.1275113113618676211013161911, 9.80391196737039086990011931292, 10.85208467445905085414239204605, 11.67728407485265433045339881508, 13.19956226006194019934470675000, 14.22395671336845618416684100881, 15.28336861073459160733032609790, 16.06355788923893206203784543529, 17.404139024569346108849736494950, 17.87652975112651804669796118113, 18.67151417273106218169817382953, 19.78816905534405980300835087105, 21.58451525329676271182574098602, 22.3597560118763575012021199330, 23.42369892027450001529459138797, 23.86050901022863247873087519448, 24.74547474494191775754775395669, 26.29488214603560656631691563983, 26.611434714115251806818186557769, 27.85841256625312547330398863186

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