Properties

Label 1-59-59.3-r0-0-0
Degree $1$
Conductor $59$
Sign $0.900 - 0.434i$
Analytic cond. $0.273994$
Root an. cond. $0.273994$
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 + (0.796 + 0.605i)3-s + (−0.161 − 0.986i)4-s + (0.468 + 0.883i)5-s + (0.976 − 0.214i)6-s + (−0.994 − 0.108i)7-s + (−0.856 − 0.515i)8-s + (0.267 + 0.963i)9-s + (0.976 + 0.214i)10-s + (−0.947 − 0.319i)11-s + (0.468 − 0.883i)12-s + (0.267 − 0.963i)13-s + (−0.725 + 0.687i)14-s + (−0.161 + 0.986i)15-s + (−0.947 + 0.319i)16-s + (−0.994 + 0.108i)17-s + ⋯
L(s)  = 1  + (0.647 − 0.762i)2-s + (0.796 + 0.605i)3-s + (−0.161 − 0.986i)4-s + (0.468 + 0.883i)5-s + (0.976 − 0.214i)6-s + (−0.994 − 0.108i)7-s + (−0.856 − 0.515i)8-s + (0.267 + 0.963i)9-s + (0.976 + 0.214i)10-s + (−0.947 − 0.319i)11-s + (0.468 − 0.883i)12-s + (0.267 − 0.963i)13-s + (−0.725 + 0.687i)14-s + (−0.161 + 0.986i)15-s + (−0.947 + 0.319i)16-s + (−0.994 + 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352081145 - 0.3088722796i\)
\(L(\frac12)\) \(\approx\) \(1.352081145 - 0.3088722796i\)
\(L(1)\) \(\approx\) \(1.475107935 - 0.2846704437i\)
\(L(1)\) \(\approx\) \(1.475107935 - 0.2846704437i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.647 - 0.762i)T \)
3 \( 1 + (0.796 + 0.605i)T \)
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

−32.69345005153809846041889790901, −31.541804126488735149027671954624, −31.21372188452783818465297325813, −29.532974793881362439590837879546, −28.73057109303876021244753388915, −26.62096856705977238963917466441, −25.75972919328933344123363181127, −24.93992682583606683240276216714, −23.988464169588034809987720692093, −22.94227880711631780354973212744, −21.32870445366448086219317142202, −20.52928382895398957238213195409, −19.01836006450341340073299258260, −17.671536681399613686641449783339, −16.33394019019890013401007110120, −15.33751942605366575251734749442, −13.759218625319049938499871222121, −13.1421064404301538753584134258, −12.18805199953256173286917142128, −9.49807509706427674197115751717, −8.504041034434956002245163189646, −7.09220124168059809875866963020, −5.8551861889986142952376880935, −4.12425293307655424069396832799, −2.42463136647401804823374474297, 2.63366714435129881037046172743, 3.30023432217150018964227261077, 5.11682617858304909781233052056, 6.74452191194648579853622609833, 8.917257850009393669946541522574, 10.26467583843641850763805368356, 10.80706496073625320257156053324, 13.03601882698775239490977695721, 13.63440298893491622912284395721, 15.036122080129688210124609026575, 15.82026868488690968847819077215, 18.080411473744379980217432670078, 19.25577853270634847594987216494, 20.152911238289662921991201335319, 21.402560996889319902677447694498, 22.17997588989722272808174602398, 23.17112685731674955762677782801, 24.893782975404639437684340916442, 26.063452695548884355808923294166, 26.94434276915863630921665576255, 28.46623758493588866434912653755, 29.46400675125528959280723193666, 30.56713868841916427476706983898, 31.44458278177787087367217760491, 32.65540145451218327720258040518

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