Properties

Label 1-59-59.5-r0-0-0
Degree $1$
Conductor $59$
Sign $0.255 + 0.966i$
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  + (0.796 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.255 + 0.966i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074766643 + 0.8280522620i\)
\(L(\frac12)\) \(\approx\) \(1.074766643 + 0.8280522620i\)
\(L(1)\) \(\approx\) \(1.300664010 + 0.6956238862i\)
\(L(1)\) \(\approx\) \(1.300664010 + 0.6956238862i\)

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

−31.77723038037340695173967755432, −31.27451151672305971108794417864, −30.55830775295752221138460367230, −29.4529564538313870579993698723, −28.37954517095579003021794302538, −26.99268240480077789892650659783, −25.511002029074424857022877600648, −24.2636383574988280666478910512, −23.52340370884705107581523737064, −22.38365976067166399760877785403, −21.02270546502061721585136434954, −19.94639543798116168933591995641, −18.755924518556228230805185005, −18.21184281302567989808975945444, −15.67127119388775924443447203969, −14.63573800028213594785461996986, −13.78414665405196854400659824963, −12.06333155852948866380513100389, −11.76515818168751467870515103193, −9.9097939666370376639493381241, −8.0149477739548723960982444274, −6.78006029252296718247719221170, −5.07577448790977239672715969536, −3.20550969829395061568446471575, −2.02991432191879575303089581860, 3.159169833529844526087624592382, 4.42730207907219585457375824037, 5.36625637119392239479570135329, 7.76110188633408091627797822329, 8.30524476382904742344090577911, 10.35051772783521970209937069277, 11.7697195766729708593708005569, 13.2588952950647857625300660931, 14.38928448297996405978577904396, 15.53319992528961270753784073368, 16.32612030542425486594809013009, 17.46797054385587873022236143632, 19.771193081296756726901452887296, 20.641117511145924698239670564216, 21.53029041802279793907613479062, 22.91412413795104183165667447794, 23.92658183340109799616951958377, 24.89858807380180109122201203866, 26.41539046497927919862137029684, 26.96704000521899530528591937926, 28.30434420534345570276427384310, 30.10080333512958519712584125100, 31.07218309983260095949519120732, 32.12379787814029314034301846812, 32.61928218914903480511556029573

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