Properties

Label 1-177-177.113-r0-0-0
Degree $1$
Conductor $177$
Sign $0.297 + 0.954i$
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  + (0.796 + 0.605i)2-s + (0.267 + 0.963i)4-s + (0.725 + 0.687i)5-s + (0.647 − 0.762i)7-s + (−0.370 + 0.928i)8-s + (0.161 + 0.986i)10-s + (−0.856 − 0.515i)11-s + (0.561 + 0.827i)13-s + (0.976 − 0.214i)14-s + (−0.856 + 0.515i)16-s + (−0.647 − 0.762i)17-s + (0.907 − 0.419i)19-s + (−0.468 + 0.883i)20-s + (−0.370 − 0.928i)22-s + (−0.947 − 0.319i)23-s + ⋯
L(s)  = 1  + (0.796 + 0.605i)2-s + (0.267 + 0.963i)4-s + (0.725 + 0.687i)5-s + (0.647 − 0.762i)7-s + (−0.370 + 0.928i)8-s + (0.161 + 0.986i)10-s + (−0.856 − 0.515i)11-s + (0.561 + 0.827i)13-s + (0.976 − 0.214i)14-s + (−0.856 + 0.515i)16-s + (−0.647 − 0.762i)17-s + (0.907 − 0.419i)19-s + (−0.468 + 0.883i)20-s + (−0.370 − 0.928i)22-s + (−0.947 − 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.297 + 0.954i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.297 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564086032 + 1.150627715i\)
\(L(\frac12)\) \(\approx\) \(1.564086032 + 1.150627715i\)
\(L(1)\) \(\approx\) \(1.546597055 + 0.7402065286i\)
\(L(1)\) \(\approx\) \(1.546597055 + 0.7402065286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.796 + 0.605i)T \)
5 \( 1 + (0.725 + 0.687i)T \)
7 \( 1 + (0.647 - 0.762i)T \)
11 \( 1 + (-0.856 - 0.515i)T \)
13 \( 1 + (0.561 + 0.827i)T \)
17 \( 1 + (-0.647 - 0.762i)T \)
19 \( 1 + (0.907 - 0.419i)T \)
23 \( 1 + (-0.947 - 0.319i)T \)
29 \( 1 + (-0.796 + 0.605i)T \)
31 \( 1 + (-0.907 - 0.419i)T \)
37 \( 1 + (0.370 + 0.928i)T \)
41 \( 1 + (0.947 - 0.319i)T \)
43 \( 1 + (0.856 - 0.515i)T \)
47 \( 1 + (-0.725 + 0.687i)T \)
53 \( 1 + (0.161 - 0.986i)T \)
61 \( 1 + (-0.796 - 0.605i)T \)
67 \( 1 + (0.370 - 0.928i)T \)
71 \( 1 + (0.725 - 0.687i)T \)
73 \( 1 + (-0.976 + 0.214i)T \)
79 \( 1 + (0.468 - 0.883i)T \)
83 \( 1 + (-0.994 + 0.108i)T \)
89 \( 1 + (0.796 - 0.605i)T \)
97 \( 1 + (-0.976 - 0.214i)T \)
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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

−27.76284379121178948829429305112, −26.06614384142498369638926532868, −24.91958733104637616908100024382, −24.35265499446640432701130135944, −23.31729608742243779100937973601, −22.19194685237209435142618719675, −21.34848019024091808037083780550, −20.6412628384336626066074285102, −19.8547900462380488446393886276, −18.28115585441712813843868105605, −17.820550454831552489390038869430, −16.09480593108631907120029884465, −15.24608067110420960745698353106, −14.16950963141662740783710045311, −13.066095098559775902835499505292, −12.49299553297250650489293239792, −11.25361451984490955420816903484, −10.20132401603516153187971555477, −9.14188403012107060025426849524, −7.82849229341786777963767748306, −5.86382534183088431695390503442, −5.43436013057407153769232785176, −4.164789914440773698677201214594, −2.52204473000475164145974957052, −1.54247581837730609467191532561, 2.111302397944854534493026334, 3.423978098135263954613152418906, 4.73827627894416490937564522101, 5.84148177181609115245080716426, 6.945214518369549699788186798454, 7.82272101997625746467022155828, 9.249824462665997397654081853680, 10.82265775030057700428076977728, 11.47148753108710715263752211906, 13.157540287022216584574250908079, 13.84074953111615115846715158760, 14.46589235133976981322194191849, 15.78120612475491845885167411092, 16.63299678157377151617095148077, 17.82298898304586012938255062167, 18.43253739825740767156172106202, 20.28362320377278545815588885052, 21.0240438130733296770411463732, 21.95148522719702457370067324375, 22.76487008332994230256557016234, 23.95605388379013594182534765126, 24.3673249088004185686183177550, 25.90769784970354391465051934513, 26.20805029145393620939731926120, 27.2073557417169856401623144424

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