Properties

Label 1-177-177.98-r0-0-0
Degree $1$
Conductor $177$
Sign $-0.990 + 0.137i$
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  + (0.647 + 0.762i)2-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (−0.994 + 0.108i)7-s + (−0.856 + 0.515i)8-s + (−0.976 + 0.214i)10-s + (−0.947 + 0.319i)11-s + (−0.267 − 0.963i)13-s + (−0.725 − 0.687i)14-s + (−0.947 − 0.319i)16-s + (0.994 + 0.108i)17-s + (0.0541 + 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.856 − 0.515i)22-s + (0.907 + 0.419i)23-s + ⋯
L(s)  = 1  + (0.647 + 0.762i)2-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (−0.994 + 0.108i)7-s + (−0.856 + 0.515i)8-s + (−0.976 + 0.214i)10-s + (−0.947 + 0.319i)11-s + (−0.267 − 0.963i)13-s + (−0.725 − 0.687i)14-s + (−0.947 − 0.319i)16-s + (0.994 + 0.108i)17-s + (0.0541 + 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.856 − 0.515i)22-s + (0.907 + 0.419i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.990 + 0.137i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ -0.990 + 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06647327469 + 0.9604672547i\)
\(L(\frac12)\) \(\approx\) \(0.06647327469 + 0.9604672547i\)
\(L(1)\) \(\approx\) \(0.7235098360 + 0.7361460856i\)
\(L(1)\) \(\approx\) \(0.7235098360 + 0.7361460856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.647 + 0.762i)T \)
5 \( 1 + (-0.468 + 0.883i)T \)
7 \( 1 + (-0.994 + 0.108i)T \)
11 \( 1 + (-0.947 + 0.319i)T \)
13 \( 1 + (-0.267 - 0.963i)T \)
17 \( 1 + (0.994 + 0.108i)T \)
19 \( 1 + (0.0541 + 0.998i)T \)
23 \( 1 + (0.907 + 0.419i)T \)
29 \( 1 + (-0.647 + 0.762i)T \)
31 \( 1 + (-0.0541 + 0.998i)T \)
37 \( 1 + (0.856 + 0.515i)T \)
41 \( 1 + (-0.907 + 0.419i)T \)
43 \( 1 + (0.947 + 0.319i)T \)
47 \( 1 + (0.468 + 0.883i)T \)
53 \( 1 + (-0.976 - 0.214i)T \)
61 \( 1 + (-0.647 - 0.762i)T \)
67 \( 1 + (0.856 - 0.515i)T \)
71 \( 1 + (-0.468 - 0.883i)T \)
73 \( 1 + (0.725 + 0.687i)T \)
79 \( 1 + (0.796 + 0.605i)T \)
83 \( 1 + (-0.370 + 0.928i)T \)
89 \( 1 + (0.647 - 0.762i)T \)
97 \( 1 + (0.725 - 0.687i)T \)
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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

−27.11415988099194240658467708446, −26.06653337522616008820021783515, −24.7063988291003358354803438503, −23.76481445530831586274821232199, −23.19115158648692357118391151287, −22.04554881532194313485797310275, −21.05687910616763248980885595728, −20.31790256234593619903569242191, −19.20765331096563822622365870138, −18.75369985229552668366150614235, −16.93701544100726559673471436700, −16.018398007407970630839891462863, −15.08194307812606851177602331077, −13.62730244187822831231640920859, −12.974266467781323384947521833482, −12.07323705765562392087887930630, −11.05272008119534632718264243540, −9.76915242997181479669688128790, −8.96251387347481626384293433947, −7.33880335361407792334545923424, −5.87873332787704165486194767000, −4.80671046076478552724899611446, −3.70261709330229062819427588437, −2.46415245603388500230125674098, −0.61401633309432513430863581370, 2.90095280743092263935957442628, 3.48391634056884322401427145349, 5.153444295481801117600313697592, 6.18554758344366703701453082091, 7.33112482630818331423548621735, 8.026927905284433854773445874218, 9.69761988194653919121364475962, 10.80731687269709927335542243270, 12.314095982053233688306614883699, 12.91006617840314338960282505618, 14.210368649395517281369552064846, 15.13741280915531115919357221127, 15.82711174964999509767226533832, 16.83439859315335814976439095226, 18.12375547809218371048219544791, 18.913914106051239167961426595144, 20.23289916503402475873682290100, 21.39945234826487790857040393564, 22.458333131009433295776109175122, 23.02996409919370874110265686303, 23.72815905580367404244279116021, 25.33258355172385618379831214112, 25.5912340126731173993432284484, 26.72026611461165497397381765686, 27.47723653904618597681162708590

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