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Properties

Label 1-59-59.32-r1-0-0
Degree $1$
Conductor $59$
Sign $-0.996 - 0.0870i$
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  + (0.856 + 0.515i)2-s + (−0.370 + 0.928i)3-s + (0.468 + 0.883i)4-s + (−0.994 − 0.108i)5-s + (−0.796 + 0.605i)6-s + (−0.947 − 0.319i)7-s + (−0.0541 + 0.998i)8-s + (−0.725 − 0.687i)9-s + (−0.796 − 0.605i)10-s + (0.561 + 0.827i)11-s + (−0.994 + 0.108i)12-s + (0.725 − 0.687i)13-s + (−0.647 − 0.762i)14-s + (0.468 − 0.883i)15-s + (−0.561 + 0.827i)16-s + (−0.947 + 0.319i)17-s + ⋯
L(s)  = 1  + (0.856 + 0.515i)2-s + (−0.370 + 0.928i)3-s + (0.468 + 0.883i)4-s + (−0.994 − 0.108i)5-s + (−0.796 + 0.605i)6-s + (−0.947 − 0.319i)7-s + (−0.0541 + 0.998i)8-s + (−0.725 − 0.687i)9-s + (−0.796 − 0.605i)10-s + (0.561 + 0.827i)11-s + (−0.994 + 0.108i)12-s + (0.725 − 0.687i)13-s + (−0.647 − 0.762i)14-s + (0.468 − 0.883i)15-s + (−0.561 + 0.827i)16-s + (−0.947 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.996 - 0.0870i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ -0.996 - 0.0870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05320709956 + 1.220016541i\)
\(L(\frac12)\) \(\approx\) \(-0.05320709956 + 1.220016541i\)
\(L(1)\) \(\approx\) \(0.7555154866 + 0.7764156431i\)
\(L(1)\) \(\approx\) \(0.7555154866 + 0.7764156431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.856 + 0.515i)T \)
3 \( 1 + (-0.370 + 0.928i)T \)
5 \( 1 + (-0.994 - 0.108i)T \)
7 \( 1 + (-0.947 - 0.319i)T \)
11 \( 1 + (0.561 + 0.827i)T \)
13 \( 1 + (0.725 - 0.687i)T \)
17 \( 1 + (-0.947 + 0.319i)T \)
19 \( 1 + (-0.161 + 0.986i)T \)
23 \( 1 + (-0.267 + 0.963i)T \)
29 \( 1 + (-0.856 + 0.515i)T \)
31 \( 1 + (0.161 + 0.986i)T \)
37 \( 1 + (-0.0541 - 0.998i)T \)
41 \( 1 + (0.267 + 0.963i)T \)
43 \( 1 + (0.561 - 0.827i)T \)
47 \( 1 + (0.994 - 0.108i)T \)
53 \( 1 + (0.796 - 0.605i)T \)
61 \( 1 + (0.856 + 0.515i)T \)
67 \( 1 + (-0.0541 + 0.998i)T \)
71 \( 1 + (-0.994 + 0.108i)T \)
73 \( 1 + (-0.647 - 0.762i)T \)
79 \( 1 + (-0.370 - 0.928i)T \)
83 \( 1 + (-0.907 - 0.419i)T \)
89 \( 1 + (0.856 - 0.515i)T \)
97 \( 1 + (-0.647 + 0.762i)T \)
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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

−31.59208664281292257535904740486, −30.77263331368107093358198135942, −29.81182470211600454315379267989, −28.75298148092398479219446509477, −27.944992363413005563612052066803, −26.14683564290855041172520707962, −24.54849020485858332154269843084, −23.90074987209572320452522793970, −22.71918757729666956194394979051, −22.106350613430951995206998194694, −20.23581311165696318984103504669, −19.209576902026307505039289032798, −18.667507327033069889357964937539, −16.53559360263013156464829934214, −15.46397614268248362872031023081, −13.862233448702887393551496091135, −12.90583674973779254465738328079, −11.72713263090613639951072579677, −11.04508254837863150227416336660, −8.87488366561585410835558632599, −6.92808787012669584953447206826, −6.072459917101450137059585682677, −4.14423850013148393872833159822, −2.627549058471976591303923626475, −0.51695089609634023128322605945, 3.49266788821337171240072202580, 4.1988933892459122396153269300, 5.81014533173642734991821490061, 7.17738505888766371641094935769, 8.82505254370860459483493711014, 10.58106278192372097161041282213, 11.86572061501920559888324104853, 12.950754050843690035258997964524, 14.67742583569185161686544856233, 15.6563344680523095775909404204, 16.31809751938766247648976668767, 17.55757760942753948367372303114, 19.78761137910361074296657399809, 20.572319069688793351195199361365, 22.09161730219262839904107351480, 22.893291161398208481487308176716, 23.511271814040815886971415289832, 25.21231282544451125895734500429, 26.236056989673847580304402117596, 27.3143807036446479292410251112, 28.49254014422706602711346664762, 29.89985823332285735982812886338, 31.167230037125328518859246269252, 32.075052739297602377726066391, 32.96546656215547614267698171710

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