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Properties

Label 1-59-59.30-r1-0-0
Degree $1$
Conductor $59$
Sign $0.998 + 0.0560i$
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  + (0.994 − 0.108i)2-s + (0.647 + 0.762i)3-s + (0.976 − 0.214i)4-s + (0.796 − 0.605i)5-s + (0.725 + 0.687i)6-s + (−0.370 − 0.928i)7-s + (0.947 − 0.319i)8-s + (−0.161 + 0.986i)9-s + (0.725 − 0.687i)10-s + (−0.907 − 0.419i)11-s + (0.796 + 0.605i)12-s + (0.161 + 0.986i)13-s + (−0.468 − 0.883i)14-s + (0.976 + 0.214i)15-s + (0.907 − 0.419i)16-s + (−0.370 + 0.928i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.108i)2-s + (0.647 + 0.762i)3-s + (0.976 − 0.214i)4-s + (0.796 − 0.605i)5-s + (0.725 + 0.687i)6-s + (−0.370 − 0.928i)7-s + (0.947 − 0.319i)8-s + (−0.161 + 0.986i)9-s + (0.725 − 0.687i)10-s + (−0.907 − 0.419i)11-s + (0.796 + 0.605i)12-s + (0.161 + 0.986i)13-s + (−0.468 − 0.883i)14-s + (0.976 + 0.214i)15-s + (0.907 − 0.419i)16-s + (−0.370 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.998 + 0.0560i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.998 + 0.0560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.580020175 + 0.1004048523i\)
\(L(\frac12)\) \(\approx\) \(3.580020175 + 0.1004048523i\)
\(L(1)\) \(\approx\) \(2.373977902 + 0.06377304803i\)
\(L(1)\) \(\approx\) \(2.373977902 + 0.06377304803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.994 - 0.108i)T \)
3 \( 1 + (0.647 + 0.762i)T \)
5 \( 1 + (0.796 - 0.605i)T \)
7 \( 1 + (-0.370 - 0.928i)T \)
11 \( 1 + (-0.907 - 0.419i)T \)
13 \( 1 + (0.161 + 0.986i)T \)
17 \( 1 + (-0.370 + 0.928i)T \)
19 \( 1 + (-0.561 + 0.827i)T \)
23 \( 1 + (-0.0541 - 0.998i)T \)
29 \( 1 + (-0.994 - 0.108i)T \)
31 \( 1 + (0.561 + 0.827i)T \)
37 \( 1 + (0.947 + 0.319i)T \)
41 \( 1 + (0.0541 - 0.998i)T \)
43 \( 1 + (-0.907 + 0.419i)T \)
47 \( 1 + (-0.796 - 0.605i)T \)
53 \( 1 + (-0.725 - 0.687i)T \)
61 \( 1 + (0.994 - 0.108i)T \)
67 \( 1 + (0.947 - 0.319i)T \)
71 \( 1 + (0.796 + 0.605i)T \)
73 \( 1 + (-0.468 - 0.883i)T \)
79 \( 1 + (0.647 - 0.762i)T \)
83 \( 1 + (0.856 + 0.515i)T \)
89 \( 1 + (0.994 + 0.108i)T \)
97 \( 1 + (-0.468 + 0.883i)T \)
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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

−32.09548154758754557457857286153, −31.430862333135279021325679177909, −30.27421884969517053194275897778, −29.535372654766663250780764453428, −28.48071257862697775070363500761, −26.17585719465830602164666039106, −25.43245406165326638452328596348, −24.73521739179535094775122734545, −23.36858043163146266898030011011, −22.285901714001723611576898542973, −21.16932553979801700209697858446, −20.06765981401385134788005612408, −18.65273526264752422915567269723, −17.64572917800839350881919651839, −15.55756829033485640229943710850, −14.839797282833665939027514764, −13.37542421142557264442682389030, −12.9167766348221083805560102275, −11.34452136985963353625035656371, −9.60463721177672086510536541144, −7.79752384304822772024104973405, −6.53568132746933082845140203280, −5.40280909695035654412179027113, −3.02060113251011306806247218426, −2.2494464169355622798629767540, 2.04116372926478101120657631602, 3.728038365217882886375818842406, 4.85208248288048885338579383266, 6.36264849652829142263036320056, 8.26295543901333977463568531525, 9.947211951421432111028949353678, 10.830732098736864329616154735653, 12.86648767692641064599680044393, 13.66232316979654464646921665444, 14.65030536090210598111459471143, 16.17897681262083353528960067640, 16.79870648934570508826282092408, 19.15322906572653677281370765051, 20.40425000739921935287006684758, 21.0762865623764176469613742294, 21.95898662061934426723830476141, 23.39871003385761082225076051006, 24.4841757550450259944175931392, 25.7462682142225584967407460017, 26.49920816650331044501129904237, 28.36132884970996936555122652225, 29.1792448275196540029184019452, 30.44759534567985868586154005390, 31.64384003072055380092752207095, 32.41267091083946281122489027031

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