Properties

Label 1-59-59.56-r1-0-0
Degree $1$
Conductor $59$
Sign $-0.871 + 0.489i$
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  + (−0.647 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.871 + 0.489i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ -0.871 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3268725709 + 1.249374812i\)
\(L(\frac12)\) \(\approx\) \(0.3268725709 + 1.249374812i\)
\(L(1)\) \(\approx\) \(0.7060556766 + 0.6883159728i\)
\(L(1)\) \(\approx\) \(0.7060556766 + 0.6883159728i\)

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

−31.934542621238848419191046159062, −30.67603383874288628214144042879, −29.52468787387764997382153463731, −28.957413397731240590797622527834, −27.58546963313497261454827755204, −26.387279497214778258780638389329, −25.211299661340871523104683174532, −24.67378497923962921305224946228, −22.66925998111074756169912494689, −21.408073816134084433073713806658, −20.05471121792166551249108189263, −19.71649057771209881465420657780, −18.33352285307464604107566707534, −17.218945212099607603551710515536, −15.93917997256474580089091996054, −13.92020060312200074582736996275, −12.85628187031796143528106235484, −12.12140801382567929941969315357, −10.02236701390910212026484687854, −9.094029344105642121786949760770, −8.09785093982188965076072609516, −6.38838178094025250015876767485, −3.87799384118786592699244967022, −2.38686363680666098786827855660, −0.802669032444658459818351810168, 2.30128744550150201276043219007, 4.21367241701027987850101752143, 6.29415089612904760546828306433, 7.27029959005301961429188816508, 9.14873209319883639395860592499, 9.66594229344762944424516043773, 10.99744940194300360711660271778, 13.5349399632591434880641896531, 14.43348497742345429469141421726, 15.47387850834109091556969204390, 16.565387489952513167615350629500, 17.88088823291300872993359063747, 19.308472592664960208470769383040, 19.832736261182619546868642400280, 21.80799708940594845125590855318, 22.58239631117645014234006186405, 24.27752158375275963582348639233, 25.5493492141813174035420592770, 26.09224802274564694343013574849, 26.89339573313098085568869017467, 28.19473130232338978463606327318, 29.436886679887358012769345320413, 30.87563025490508879000525737949, 32.20270844375283533649991731756, 33.04218391567308082284788620790

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