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Properties

Label 1-588-588.191-r0-0-0
Degree $1$
Conductor $588$
Sign $0.747 + 0.664i$
Analytic cond. $2.73066$
Root an. cond. $2.73066$
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  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.623 + 0.781i)41-s + (−0.623 − 0.781i)43-s + (0.955 − 0.294i)47-s + (−0.826 + 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.623 + 0.781i)41-s + (−0.623 − 0.781i)43-s + (0.955 − 0.294i)47-s + (−0.826 + 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(2.73066\)
Root analytic conductor: \(2.73066\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (0:\ ),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469382491 + 0.5586452201i\)
\(L(\frac12)\) \(\approx\) \(1.469382491 + 0.5586452201i\)
\(L(1)\) \(\approx\) \(1.209847455 + 0.1732642027i\)
\(L(1)\) \(\approx\) \(1.209847455 + 0.1732642027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + T \)
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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

−22.95794049097541166573368523313, −22.2293524836475642317365271930, −21.42115393781468105481590607293, −20.46445472185257953871255300398, −20.08825309931704698480750093427, −18.713209734693878109580035825969, −17.8419757702384167970612272322, −17.55645447147290610213221730780, −16.30595258155616634325123796560, −15.58712078520363968735755478011, −14.5542321902003195179466707630, −13.72286249116270762156395783369, −12.913199982728020891640206141201, −12.21652917065589795225709767272, −10.88740045052093209143115561288, −10.14829043397120883383594536147, −9.39802051954510056705736708237, −8.38953168571429067970614166600, −7.30130460920537774603582201483, −6.42544562389186772485370733749, −5.190397054883865806206202315745, −4.79575823939295237326733543314, −2.98524312490643046108239975057, −2.34980503826450082432990464611, −0.87153106202095732045513466697, 1.36759322924141646382438038894, 2.37228801095248370950758478848, 3.474989908375281565541133757387, 4.737109221092031391516195510088, 5.80390900983590619424398958387, 6.39496459486492578129302510135, 7.64709209266289242790667754417, 8.57535857609234339854522902477, 9.632547038440670217041111224497, 10.26281659825728222771891144404, 11.2575889107041726517611090122, 12.21129058655063212029384013892, 13.37641390628385667757976173052, 13.78050628729807142262811729122, 14.76254870722219198971974930579, 15.71872667587043615777492792210, 16.77654947325434208985216834899, 17.27018233945340731516879359885, 18.429875466295720946584411839593, 18.868407139946885201967779510915, 19.97368152357600396414416748679, 21.051821838150150437819164244294, 21.525952279504667589642338973970, 22.2027508590677567522135632046, 23.4046221711791962608637613263

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