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Properties

Label 1-177-177.131-r0-0-0
Degree $1$
Conductor $177$
Sign $0.186 + 0.982i$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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  + (−0.161 + 0.986i)2-s + (−0.947 − 0.319i)4-s + (0.561 + 0.827i)5-s + (0.976 − 0.214i)7-s + (0.468 − 0.883i)8-s + (−0.907 + 0.419i)10-s + (0.796 − 0.605i)11-s + (0.856 − 0.515i)13-s + (0.0541 + 0.998i)14-s + (0.796 + 0.605i)16-s + (−0.976 − 0.214i)17-s + (−0.994 + 0.108i)19-s + (−0.267 − 0.963i)20-s + (0.468 + 0.883i)22-s + (0.647 + 0.762i)23-s + ⋯
L(s)  = 1  + (−0.161 + 0.986i)2-s + (−0.947 − 0.319i)4-s + (0.561 + 0.827i)5-s + (0.976 − 0.214i)7-s + (0.468 − 0.883i)8-s + (−0.907 + 0.419i)10-s + (0.796 − 0.605i)11-s + (0.856 − 0.515i)13-s + (0.0541 + 0.998i)14-s + (0.796 + 0.605i)16-s + (−0.976 − 0.214i)17-s + (−0.994 + 0.108i)19-s + (−0.267 − 0.963i)20-s + (0.468 + 0.883i)22-s + (0.647 + 0.762i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.186 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9157505349 + 0.7581257033i\)
\(L(\frac12)\) \(\approx\) \(0.9157505349 + 0.7581257033i\)
\(L(1)\) \(\approx\) \(0.9553664522 + 0.5356702397i\)
\(L(1)\) \(\approx\) \(0.9553664522 + 0.5356702397i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.161 + 0.986i)T \)
5 \( 1 + (0.561 + 0.827i)T \)
7 \( 1 + (0.976 - 0.214i)T \)
11 \( 1 + (0.796 - 0.605i)T \)
13 \( 1 + (0.856 - 0.515i)T \)
17 \( 1 + (-0.976 - 0.214i)T \)
19 \( 1 + (-0.994 + 0.108i)T \)
23 \( 1 + (0.647 + 0.762i)T \)
29 \( 1 + (0.161 + 0.986i)T \)
31 \( 1 + (0.994 + 0.108i)T \)
37 \( 1 + (-0.468 - 0.883i)T \)
41 \( 1 + (-0.647 + 0.762i)T \)
43 \( 1 + (-0.796 - 0.605i)T \)
47 \( 1 + (-0.561 + 0.827i)T \)
53 \( 1 + (-0.907 - 0.419i)T \)
61 \( 1 + (0.161 - 0.986i)T \)
67 \( 1 + (-0.468 + 0.883i)T \)
71 \( 1 + (0.561 - 0.827i)T \)
73 \( 1 + (-0.0541 - 0.998i)T \)
79 \( 1 + (0.267 + 0.963i)T \)
83 \( 1 + (-0.725 - 0.687i)T \)
89 \( 1 + (-0.161 - 0.986i)T \)
97 \( 1 + (-0.0541 + 0.998i)T \)
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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

−27.55294127766922472806144229501, −26.43041767300408416281806453184, −25.2801534670919860248703358237, −24.31472127643269179988587238620, −23.23268568967414709288339608498, −22.06586739013878877602103228822, −21.11095350538836396224647543493, −20.63840092133079790751180734353, −19.580798003069032672905766637350, −18.45034276531549674612957409370, −17.429847845640995936980856955, −16.93611962708596998129327692589, −15.200395605496739261951911682174, −13.9903638402873381646154268597, −13.14959919020559845253259205834, −12.07324317783673656246396239080, −11.22995123163425310468548391869, −10.05473010875572344733331070657, −8.832462076966356569595464458522, −8.40800726502376885623005846538, −6.42416721190768756487079342573, −4.83021311508414840362965851722, −4.19989413415300380866902625888, −2.22311651746448257792913378436, −1.359730937241688217745999227211, 1.51359228918604729666927175021, 3.50999367113952131980064224344, 4.8925573274978949982511946914, 6.12875307672999432213595832939, 6.90874473819873212914299197367, 8.21473132640096527489622020058, 9.08498056626262069311890878536, 10.49053569607751262372151030634, 11.2787723170204378549590432826, 13.17349419436738950468207970194, 13.999857988531275465941886196619, 14.76860826250264963738640215090, 15.69537954253220576859337928092, 17.06887699649769468486293319774, 17.68807261333257239243681196931, 18.51971011852208898757471683773, 19.58434046886070133815731615961, 21.13025077337325467545364427833, 21.990454812380975935790702677785, 22.96296049042815355740808109649, 23.86265585535540725495520532506, 24.930814218497387186647776740429, 25.52382421592975946476925187951, 26.70489804951389228565679617913, 27.241032316132522932969723567000

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