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Properties

Label 1-177-177.23-r0-0-0
Degree $1$
Conductor $177$
Sign $-0.336 - 0.941i$
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  + (0.0541 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ -0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6631636711 - 0.9407974654i\)
\(L(\frac12)\) \(\approx\) \(0.6631636711 - 0.9407974654i\)
\(L(1)\) \(\approx\) \(0.8846707007 - 0.6308501493i\)
\(L(1)\) \(\approx\) \(0.8846707007 - 0.6308501493i\)

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

−27.66354394475423261922995158614, −26.415928476621050464285973793355, −25.58562059985232620764794991990, −24.87927014582651241357087144929, −24.20210287690864512092465623748, −22.868808285919352639787834575751, −21.95707299868613562571462695006, −21.4164319547219020723572684082, −19.76291289537601730365018878531, −18.66780706083751551527188139753, −17.77965215730076318890420103596, −16.71643778470129964983761372874, −16.18616940981944284238140972688, −14.59216456186240858919088582177, −14.273404078656267394469524831212, −12.77983752993217228563028768653, −12.187713756154614147177309260199, −10.056328118374813103798357316229, −9.325562376687566874455188425847, −8.45260996303916722562181132896, −6.87583174482950886253167083777, −6.04967436009956594085515122379, −5.11024281155413531117384309068, −3.66800960699092084052453806020, −1.785303693583348462323872389666, 1.02361980548727528083979441062, 2.58041747458583184565104875854, 3.61899498087861079847924340979, 5.04076198699323811315512781791, 6.29520945649765306903501610208, 7.70326209563878273764015416114, 9.53519824907105966250168064325, 9.73571266656276063539189725772, 10.98360088834670722776646231629, 12.03117104836599559084711053376, 13.35010208498437689184481057298, 13.79860937665468132019451062812, 14.91881800611455715081917984838, 16.713728657878517346943502820425, 17.47133293914144970772883019007, 18.40485968441452287692183216582, 19.60817123731892147481649327797, 20.206288303467860974522019976418, 21.338816606012958168517585418093, 22.324008176709249005487756105727, 22.73753392618562258626400663118, 24.141217189733617031639500250950, 25.3456562116178170736440870011, 26.34553596513215936243719269778, 27.18059147111398693104289671872

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