Properties

Label 1-177-177.95-r1-0-0
Degree $1$
Conductor $177$
Sign $0.819 + 0.572i$
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  + (−0.0541 + 0.998i)2-s + (−0.994 − 0.108i)4-s + (0.947 + 0.319i)5-s + (−0.561 − 0.827i)7-s + (0.161 − 0.986i)8-s + (−0.370 + 0.928i)10-s + (−0.976 + 0.214i)11-s + (0.647 + 0.762i)13-s + (0.856 − 0.515i)14-s + (0.976 + 0.214i)16-s + (0.561 − 0.827i)17-s + (0.468 − 0.883i)19-s + (−0.907 − 0.419i)20-s + (−0.161 − 0.986i)22-s + (0.725 − 0.687i)23-s + ⋯
L(s)  = 1  + (−0.0541 + 0.998i)2-s + (−0.994 − 0.108i)4-s + (0.947 + 0.319i)5-s + (−0.561 − 0.827i)7-s + (0.161 − 0.986i)8-s + (−0.370 + 0.928i)10-s + (−0.976 + 0.214i)11-s + (0.647 + 0.762i)13-s + (0.856 − 0.515i)14-s + (0.976 + 0.214i)16-s + (0.561 − 0.827i)17-s + (0.468 − 0.883i)19-s + (−0.907 − 0.419i)20-s + (−0.161 − 0.986i)22-s + (0.725 − 0.687i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647412338 + 0.5187503121i\)
\(L(\frac12)\) \(\approx\) \(1.647412338 + 0.5187503121i\)
\(L(1)\) \(\approx\) \(1.037798141 + 0.3863170399i\)
\(L(1)\) \(\approx\) \(1.037798141 + 0.3863170399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.0541 + 0.998i)T \)
5 \( 1 + (0.947 + 0.319i)T \)
7 \( 1 + (-0.561 - 0.827i)T \)
11 \( 1 + (-0.976 + 0.214i)T \)
13 \( 1 + (0.647 + 0.762i)T \)
17 \( 1 + (0.561 - 0.827i)T \)
19 \( 1 + (0.468 - 0.883i)T \)
23 \( 1 + (0.725 - 0.687i)T \)
29 \( 1 + (-0.0541 - 0.998i)T \)
31 \( 1 + (0.468 + 0.883i)T \)
37 \( 1 + (-0.161 - 0.986i)T \)
41 \( 1 + (0.725 + 0.687i)T \)
43 \( 1 + (0.976 + 0.214i)T \)
47 \( 1 + (0.947 - 0.319i)T \)
53 \( 1 + (0.370 + 0.928i)T \)
61 \( 1 + (0.0541 - 0.998i)T \)
67 \( 1 + (-0.161 + 0.986i)T \)
71 \( 1 + (0.947 - 0.319i)T \)
73 \( 1 + (-0.856 + 0.515i)T \)
79 \( 1 + (0.907 + 0.419i)T \)
83 \( 1 + (-0.267 - 0.963i)T \)
89 \( 1 + (-0.0541 - 0.998i)T \)
97 \( 1 + (-0.856 - 0.515i)T \)
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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

−27.39973435215108186055065143534, −25.9368062707933518966688832983, −25.50626226984917573865984239946, −24.10356516437702977005034717009, −22.90294172011715327691551409432, −22.05548110740039456618112638741, −21.097231115406807721866949982, −20.61011021220076436100984787686, −19.19070366061863510834469149367, −18.44650500301697498382710795872, −17.61041053081062272869588486252, −16.43358576935715382865797872014, −15.1254851727008155932930428399, −13.766170281849462102200805028, −12.95506066362416465692313789895, −12.2810279796470282350276984800, −10.789558466022080434691965680807, −9.99424076289141674682608633686, −9.01527991111342489965698465323, −8.02353412847689860796257308134, −5.86367468134462043512801499514, −5.30882125172331822740891567781, −3.47714003103660510612426525483, −2.455899100749015933660503903942, −1.10080214561281664233250865436, 0.8004824197256543941074912298, 2.86565696498429127491231630723, 4.45226696062326705926166130663, 5.62539803060932917100453396867, 6.71822727634860326299332381385, 7.4783855729390963480723690374, 9.01400257145245294448715420091, 9.85425143784610641585241416208, 10.84085099728091203711116774470, 12.78679091820615795140271859521, 13.65194883930801541757097739695, 14.21142831163283926236965770303, 15.63041913193492557603509987313, 16.42446126385154819411088035363, 17.39888842968819961836488280186, 18.26191269255600331285908452248, 19.12562329726396756557890216752, 20.67695594945930832977262273875, 21.5836548768010132510490958150, 22.84000994708396986087608972569, 23.294219254389050000877238805, 24.52912927222786721089394564138, 25.42335890134529983784571545983, 26.37158883904620569382715020936, 26.59714593093942694142276266450

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