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Properties

Label 1-179-179.49-r0-0-0
Degree $1$
Conductor $179$
Sign $0.483 - 0.875i$
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  + (0.880 − 0.474i)2-s + (−0.999 − 0.0352i)3-s + (0.550 − 0.835i)4-s + (0.607 + 0.794i)5-s + (−0.896 + 0.442i)6-s + (−0.949 − 0.312i)7-s + (0.0881 − 0.996i)8-s + (0.997 + 0.0705i)9-s + (0.911 + 0.411i)10-s + (0.427 − 0.904i)11-s + (−0.579 + 0.815i)12-s + (0.977 + 0.210i)13-s + (−0.984 + 0.175i)14-s + (−0.579 − 0.815i)15-s + (−0.394 − 0.918i)16-s + (0.938 − 0.345i)17-s + ⋯
L(s)  = 1  + (0.880 − 0.474i)2-s + (−0.999 − 0.0352i)3-s + (0.550 − 0.835i)4-s + (0.607 + 0.794i)5-s + (−0.896 + 0.442i)6-s + (−0.949 − 0.312i)7-s + (0.0881 − 0.996i)8-s + (0.997 + 0.0705i)9-s + (0.911 + 0.411i)10-s + (0.427 − 0.904i)11-s + (−0.579 + 0.815i)12-s + (0.977 + 0.210i)13-s + (−0.984 + 0.175i)14-s + (−0.579 − 0.815i)15-s + (−0.394 − 0.918i)16-s + (0.938 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.483 - 0.875i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.483 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279681912 - 0.7552764265i\)
\(L(\frac12)\) \(\approx\) \(1.279681912 - 0.7552764265i\)
\(L(1)\) \(\approx\) \(1.282139473 - 0.4563975853i\)
\(L(1)\) \(\approx\) \(1.282139473 - 0.4563975853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.880 - 0.474i)T \)
3 \( 1 + (-0.999 - 0.0352i)T \)
5 \( 1 + (0.607 + 0.794i)T \)
7 \( 1 + (-0.949 - 0.312i)T \)
11 \( 1 + (0.427 - 0.904i)T \)
13 \( 1 + (0.977 + 0.210i)T \)
17 \( 1 + (0.938 - 0.345i)T \)
19 \( 1 + (0.0176 - 0.999i)T \)
23 \( 1 + (-0.737 + 0.675i)T \)
29 \( 1 + (-0.192 - 0.981i)T \)
31 \( 1 + (0.713 + 0.700i)T \)
37 \( 1 + (-0.192 + 0.981i)T \)
41 \( 1 + (0.362 - 0.932i)T \)
43 \( 1 + (-0.261 - 0.965i)T \)
47 \( 1 + (0.489 + 0.871i)T \)
53 \( 1 + (-0.994 + 0.105i)T \)
59 \( 1 + (-0.863 + 0.505i)T \)
61 \( 1 + (-0.520 + 0.854i)T \)
67 \( 1 + (0.0881 + 0.996i)T \)
71 \( 1 + (-0.635 + 0.772i)T \)
73 \( 1 + (-0.825 + 0.564i)T \)
79 \( 1 + (0.662 + 0.749i)T \)
83 \( 1 + (0.760 + 0.648i)T \)
89 \( 1 + (0.880 + 0.474i)T \)
97 \( 1 + (-0.123 - 0.992i)T \)
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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

−27.93016845322654592022194991187, −26.20977529806976550052176396130, −25.28936544028973288168413841155, −24.660936127123971826419850525319, −23.39918232453097969046023510490, −22.87827507485566275081110476823, −21.94895994890222625096358055826, −21.073475694785214069393083749373, −20.16091216371204474423781060380, −18.48220995510029212555415710306, −17.43317661242221070752066132039, −16.46112182191382883321698235225, −16.07271480428938175074541198587, −14.75086125824191415339536064543, −13.41144256458138483929535550420, −12.49300173536985005049843041939, −12.13230114326557728460754254881, −10.51207705194656156639926662513, −9.41096240544435079567817672468, −7.88663363454386804165612974685, −6.33747459602072306454524316592, −5.95866351646392520661895321300, −4.77734396329498381284646000138, −3.63928932147254755104942682406, −1.69486853908868891322685905647, 1.194445405314730800575069082671, 2.96196560414187390565731538472, 3.99236570205287485982436595639, 5.63525823034161079176019824064, 6.23204540432011682512638384104, 7.09195367465139032128823462423, 9.505784262673119604363531172079, 10.398236506415842035956074696238, 11.22324383320960344184382360570, 12.129528281897372260852600961545, 13.50927802989282925068139331655, 13.82028701336514927167644653264, 15.474436633523473119940144574353, 16.23195192421515735381162731362, 17.407226270054560841502746810511, 18.73862896985643164729905101237, 19.22368510698833416967290001181, 20.7987986757898301321478422955, 21.75515786258052986125530082715, 22.32541837251885348974423872282, 23.14691971985920630541445331057, 23.90259213202578826074997143352, 25.13458957270945349628719355472, 26.143339873553087885479847336344, 27.45390299208871804377832680536

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