Properties

Label 1-177-177.122-r1-0-0
Degree $1$
Conductor $177$
Sign $0.324 + 0.945i$
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  + (−0.976 − 0.214i)2-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (−0.725 − 0.687i)7-s + (−0.796 − 0.605i)8-s + (0.0541 + 0.998i)10-s + (−0.647 + 0.762i)11-s + (−0.947 − 0.319i)13-s + (0.561 + 0.827i)14-s + (0.647 + 0.762i)16-s + (0.725 − 0.687i)17-s + (−0.370 + 0.928i)19-s + (0.161 − 0.986i)20-s + (0.796 − 0.605i)22-s + (0.994 + 0.108i)23-s + ⋯
L(s)  = 1  + (−0.976 − 0.214i)2-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (−0.725 − 0.687i)7-s + (−0.796 − 0.605i)8-s + (0.0541 + 0.998i)10-s + (−0.647 + 0.762i)11-s + (−0.947 − 0.319i)13-s + (0.561 + 0.827i)14-s + (0.647 + 0.762i)16-s + (0.725 − 0.687i)17-s + (−0.370 + 0.928i)19-s + (0.161 − 0.986i)20-s + (0.796 − 0.605i)22-s + (0.994 + 0.108i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.324 + 0.945i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.324 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2790386478 + 0.1992912974i\)
\(L(\frac12)\) \(\approx\) \(0.2790386478 + 0.1992912974i\)
\(L(1)\) \(\approx\) \(0.5048316767 - 0.1060353904i\)
\(L(1)\) \(\approx\) \(0.5048316767 - 0.1060353904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.976 - 0.214i)T \)
5 \( 1 + (-0.267 - 0.963i)T \)
7 \( 1 + (-0.725 - 0.687i)T \)
11 \( 1 + (-0.647 + 0.762i)T \)
13 \( 1 + (-0.947 - 0.319i)T \)
17 \( 1 + (0.725 - 0.687i)T \)
19 \( 1 + (-0.370 + 0.928i)T \)
23 \( 1 + (0.994 + 0.108i)T \)
29 \( 1 + (-0.976 + 0.214i)T \)
31 \( 1 + (-0.370 - 0.928i)T \)
37 \( 1 + (0.796 - 0.605i)T \)
41 \( 1 + (0.994 - 0.108i)T \)
43 \( 1 + (0.647 + 0.762i)T \)
47 \( 1 + (-0.267 + 0.963i)T \)
53 \( 1 + (-0.0541 + 0.998i)T \)
61 \( 1 + (0.976 + 0.214i)T \)
67 \( 1 + (0.796 + 0.605i)T \)
71 \( 1 + (-0.267 + 0.963i)T \)
73 \( 1 + (-0.561 - 0.827i)T \)
79 \( 1 + (-0.161 + 0.986i)T \)
83 \( 1 + (-0.468 + 0.883i)T \)
89 \( 1 + (-0.976 + 0.214i)T \)
97 \( 1 + (-0.561 + 0.827i)T \)
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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

−26.798522520302675915547155000040, −26.13807574264683353188008934294, −25.379367238822299172963896795525, −24.22947496715744904555533106564, −23.31362199050619266355195862162, −22.0160910873095232024767017186, −21.24622625107934831300519777598, −19.64503701585230258401855175637, −19.06839371945690446598500674101, −18.43084860983335251426504184155, −17.226422699083449529113685613520, −16.20528214211392337149247710165, −15.25492786197939215004924096651, −14.54010205559738684214912206705, −12.88033183583618557910856013715, −11.61829264643758869074603926804, −10.710649686525685969452256741720, −9.73886173185334656910827805224, −8.661573549074372900278945206431, −7.46851624602290861662222153930, −6.55829419217962331486487794342, −5.474381651345547010969926150618, −3.22730666533434033764474709390, −2.335599855290623458780603174647, −0.19872000300681482267629863163, 1.01335659255534354784310977720, 2.632946019209970441519053866263, 4.10649325895933709452072538680, 5.62275287446514629462243311444, 7.30951937224293620353740655779, 7.81367349600835803597377503091, 9.36224275757891009913031313698, 9.87997188631503922928073525364, 11.12893719155170488749084059338, 12.51311942286220992871978164013, 12.88645117870991250840333129591, 14.7595463600124081381628334792, 15.96177236752247099805166747466, 16.69303736421210522219513280666, 17.4174815092865073254912725029, 18.71832141051004104276453271622, 19.609735481322490416772647436140, 20.458851093045230101675399388983, 21.03181027559844559146783628787, 22.63145010849241701512626361124, 23.61300856578236745044611191382, 24.7687826926728800475157764071, 25.4950638706292895458985587186, 26.52199234526357289503660161781, 27.43001199173083707453870603980

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