Properties

Label 1-59-59.9-r0-0-0
Degree $1$
Conductor $59$
Sign $0.901 + 0.432i$
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  + (−0.161 − 0.986i)2-s + (0.267 + 0.963i)3-s + (−0.947 + 0.319i)4-s + (−0.561 + 0.827i)5-s + (0.907 − 0.419i)6-s + (0.976 + 0.214i)7-s + (0.468 + 0.883i)8-s + (−0.856 + 0.515i)9-s + (0.907 + 0.419i)10-s + (0.796 + 0.605i)11-s + (−0.561 − 0.827i)12-s + (−0.856 − 0.515i)13-s + (0.0541 − 0.998i)14-s + (−0.947 − 0.319i)15-s + (0.796 − 0.605i)16-s + (0.976 − 0.214i)17-s + ⋯
L(s)  = 1  + (−0.161 − 0.986i)2-s + (0.267 + 0.963i)3-s + (−0.947 + 0.319i)4-s + (−0.561 + 0.827i)5-s + (0.907 − 0.419i)6-s + (0.976 + 0.214i)7-s + (0.468 + 0.883i)8-s + (−0.856 + 0.515i)9-s + (0.907 + 0.419i)10-s + (0.796 + 0.605i)11-s + (−0.561 − 0.827i)12-s + (−0.856 − 0.515i)13-s + (0.0541 − 0.998i)14-s + (−0.947 − 0.319i)15-s + (0.796 − 0.605i)16-s + (0.976 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.901 + 0.432i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.901 + 0.432i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7788700678 + 0.1771731835i\)
\(L(\frac12)\) \(\approx\) \(0.7788700678 + 0.1771731835i\)
\(L(1)\) \(\approx\) \(0.9105464642 + 0.04848815178i\)
\(L(1)\) \(\approx\) \(0.9105464642 + 0.04848815178i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.161 - 0.986i)T \)
3 \( 1 + (0.267 + 0.963i)T \)
5 \( 1 + (-0.561 + 0.827i)T \)
7 \( 1 + (0.976 + 0.214i)T \)
11 \( 1 + (0.796 + 0.605i)T \)
13 \( 1 + (-0.856 - 0.515i)T \)
17 \( 1 + (0.976 - 0.214i)T \)
19 \( 1 + (-0.994 - 0.108i)T \)
23 \( 1 + (0.647 - 0.762i)T \)
29 \( 1 + (-0.161 + 0.986i)T \)
31 \( 1 + (-0.994 + 0.108i)T \)
37 \( 1 + (0.468 - 0.883i)T \)
41 \( 1 + (0.647 + 0.762i)T \)
43 \( 1 + (0.796 - 0.605i)T \)
47 \( 1 + (-0.561 - 0.827i)T \)
53 \( 1 + (0.907 - 0.419i)T \)
61 \( 1 + (-0.161 - 0.986i)T \)
67 \( 1 + (0.468 + 0.883i)T \)
71 \( 1 + (-0.561 - 0.827i)T \)
73 \( 1 + (0.0541 - 0.998i)T \)
79 \( 1 + (0.267 - 0.963i)T \)
83 \( 1 + (-0.725 + 0.687i)T \)
89 \( 1 + (-0.161 + 0.986i)T \)
97 \( 1 + (0.0541 + 0.998i)T \)
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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

−32.512505823543241356803786904762, −31.6594104770802139185698438063, −30.71908713580891784307648003823, −29.37009933931591986339666316253, −27.810459357345034338932983665366, −27.10553797446871264538003423603, −25.622119303367069739227338680588, −24.50094737112464591876477383353, −24.04609989679728088233072430193, −23.08329476930566949676256524835, −21.259130251081512381937260567580, −19.63353585004063153976672699182, −18.87919477043346747835759983656, −17.299794323974057227143691293605, −16.78581992844331907705895169127, −14.95647192633513116319843635937, −14.11460147587998532926253175222, −12.79584080619215702716363733249, −11.54591315108665122580163712661, −9.19332522206094118193750726999, −8.16384987885669460075132687874, −7.29515828888280951698953612417, −5.693549218175168299084904597150, −4.16438009712193320289016708134, −1.25825758017440830168087414108, 2.4255588626000041062379474363, 3.82875335344363583887463059559, 5.011936696130559543866111458941, 7.66597626972121917271504633036, 8.9849146178034643159555162788, 10.32033132782204160633376664667, 11.179596657456333621091706876648, 12.34097471599838367405789455263, 14.6368693622850167097415457076, 14.66213977116039953193352952527, 16.79457041951002497316265317832, 18.001927525898670879118018179615, 19.347146549106594584978827308836, 20.26494075853000325772840714877, 21.408284375898842809518505343163, 22.2869171514293496400762555316, 23.26515197777115316303325481667, 25.29420135508598035576402889067, 26.57315610665893355754309458573, 27.53128137582715314093108642223, 27.82967646970651119573473258624, 29.65948654057614760263644936542, 30.63633977933741748038715468662, 31.433610782225745378616898235, 32.556959282338099621442573924422

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