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Properties

Label 1-59-59.54-r1-0-0
Degree $1$
Conductor $59$
Sign $0.982 + 0.184i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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 + (0.468 + 0.883i)3-s + (0.267 + 0.963i)4-s + (−0.725 − 0.687i)5-s + (0.161 − 0.986i)6-s + (0.647 − 0.762i)7-s + (0.370 − 0.928i)8-s + (−0.561 + 0.827i)9-s + (0.161 + 0.986i)10-s + (0.856 + 0.515i)11-s + (−0.725 + 0.687i)12-s + (0.561 + 0.827i)13-s + (−0.976 + 0.214i)14-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (0.647 + 0.762i)17-s + ⋯
L(s)  = 1  + (−0.796 − 0.605i)2-s + (0.468 + 0.883i)3-s + (0.267 + 0.963i)4-s + (−0.725 − 0.687i)5-s + (0.161 − 0.986i)6-s + (0.647 − 0.762i)7-s + (0.370 − 0.928i)8-s + (−0.561 + 0.827i)9-s + (0.161 + 0.986i)10-s + (0.856 + 0.515i)11-s + (−0.725 + 0.687i)12-s + (0.561 + 0.827i)13-s + (−0.976 + 0.214i)14-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (0.647 + 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.982 + 0.184i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.982 + 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243157174 + 0.1154502987i\)
\(L(\frac12)\) \(\approx\) \(1.243157174 + 0.1154502987i\)
\(L(1)\) \(\approx\) \(0.9234326712 + 0.001765433699i\)
\(L(1)\) \(\approx\) \(0.9234326712 + 0.001765433699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.796 - 0.605i)T \)
3 \( 1 + (0.468 + 0.883i)T \)
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

−32.43829077908576615078236532786, −31.202058951919969541683340515557, −30.24238627088296364903862308853, −29.06510218186801103743985939153, −27.56042684251466632854348248782, −26.924989191152685244978276435574, −25.429915950595552003465624522497, −24.844316404092269236235374722341, −23.70640786869550535921981899394, −22.62745744854019431168862039240, −20.54525961559308523536954596787, −19.3960682588700502183578475983, −18.51420617811747966198338440953, −17.830421599789700748403125175462, −16.105136197628839834402623431163, −14.8296387128389219559093613134, −14.13553992163591479157934333531, −12.06305836021564123716920125095, −11.00277603750936913845099823222, −9.06313678467817824424563070901, −8.06249564611507346634718284704, −7.06394186087621548500012961448, −5.69640839344101116160104848266, −3.002996164829937216143783497644, −1.07647421175736545053446865249, 1.34416088849346042729285381016, 3.60969626598638328439443972966, 4.52653110236730743981851654753, 7.41650433755093948030742733059, 8.56573719615168478330390369429, 9.558501931133644822072365623973, 10.95818369385396316070969218149, 11.907053605991672748140135636796, 13.65040072555726812152213743702, 15.21075431344987345365509988611, 16.53432035637479290046388577574, 17.199657906579202537830938661737, 19.06404033146562277556635275178, 20.0941085136165178728968827747, 20.694796553945096913432455317726, 21.815117168862945472807569585215, 23.39427620622614863486363137828, 24.93404459760400216167635320148, 26.15914845386328922108310080412, 27.151610673210869169923695452752, 27.77429862176272870487242204245, 28.76281670126170554629545881774, 30.56852141317963876483741886501, 30.985520243684440856195584019518, 32.52427391296762006217515711585

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