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Properties

Label 1-3311-3311.488-r1-0-0
Degree $1$
Conductor $3311$
Sign $-0.599 + 0.800i$
Analytic cond. $355.816$
Root an. cond. $355.816$
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  + (−0.873 − 0.486i)2-s + (0.946 + 0.323i)3-s + (0.525 + 0.850i)4-s + (−0.936 + 0.351i)5-s + (−0.669 − 0.743i)6-s + (−0.0448 − 0.998i)8-s + (0.791 + 0.611i)9-s + (0.988 + 0.149i)10-s + (0.222 + 0.974i)12-s + (−0.998 − 0.0598i)13-s + (−0.999 + 0.0299i)15-s + (−0.447 + 0.894i)16-s + (0.550 + 0.834i)17-s + (−0.393 − 0.919i)18-s + (−0.983 − 0.178i)19-s + (−0.791 − 0.611i)20-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)2-s + (0.946 + 0.323i)3-s + (0.525 + 0.850i)4-s + (−0.936 + 0.351i)5-s + (−0.669 − 0.743i)6-s + (−0.0448 − 0.998i)8-s + (0.791 + 0.611i)9-s + (0.988 + 0.149i)10-s + (0.222 + 0.974i)12-s + (−0.998 − 0.0598i)13-s + (−0.999 + 0.0299i)15-s + (−0.447 + 0.894i)16-s + (0.550 + 0.834i)17-s + (−0.393 − 0.919i)18-s + (−0.983 − 0.178i)19-s + (−0.791 − 0.611i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ -0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3933808364 + 0.7860622747i\)
\(L(\frac12)\) \(\approx\) \(0.3933808364 + 0.7860622747i\)
\(L(1)\) \(\approx\) \(0.7690628668 + 0.06938223388i\)
\(L(1)\) \(\approx\) \(0.7690628668 + 0.06938223388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.873 - 0.486i)T \)
3 \( 1 + (0.946 + 0.323i)T \)
5 \( 1 + (-0.936 + 0.351i)T \)
13 \( 1 + (-0.998 - 0.0598i)T \)
17 \( 1 + (0.550 + 0.834i)T \)
19 \( 1 + (-0.983 - 0.178i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.193 - 0.981i)T \)
31 \( 1 + (0.873 + 0.486i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.393 - 0.919i)T \)
47 \( 1 + (0.337 + 0.941i)T \)
53 \( 1 + (-0.163 - 0.986i)T \)
59 \( 1 + (0.842 + 0.538i)T \)
61 \( 1 + (0.873 - 0.486i)T \)
67 \( 1 + (0.826 + 0.563i)T \)
71 \( 1 + (0.791 - 0.611i)T \)
73 \( 1 + (0.963 + 0.266i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.873 - 0.486i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.134 + 0.990i)T \)
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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

−18.587302215721156926340005234310, −17.86256557376569394744910900211, −17.01393597687340337615243374778, −16.34175057945139608927048091200, −15.65404865538572506386778528827, −15.11421291602912520113334981044, −14.41605896745452461335484467888, −13.897715888893088264150586905271, −12.74327260471796785128020717076, −12.14339655710757413292105864718, −11.47140598296900950044040651403, −10.43415816310178447287886386626, −9.731186921030672085656326322, −9.12382586267975862792895443848, −8.288125191411326164506303697675, −7.938688755844039306943174554986, −7.16508930645809315102990206750, −6.693789632897019419505293642343, −5.48641118721470934295903729028, −4.64594528086816984161546349473, −3.76759270128176057431091836464, −2.761664890551038870478454289767, −2.02737598778900348783827866205, −1.014872496088684959237507306082, −0.21066397763515388115528761994, 0.82223445902078898253998102794, 2.10733425244487328042506284592, 2.53456647627635589297728492677, 3.52180580814191197935614958346, 4.018702507855798648750400154692, 4.820413792262768799587301550456, 6.364780674088720508381127363907, 7.07505717525928970456433789860, 7.91563156959391229941791474934, 8.21716759387349009927233180659, 8.91437669689325291189983769672, 9.95217004499903937257506179398, 10.26081208842948835287765568148, 10.99642065977676632546736676891, 11.98735089037287882724920736668, 12.415369412906375203550526229459, 13.21839969855252181668724624367, 14.31929104483700904627184836652, 14.80834798639667853022660890282, 15.677159422069257886820754904199, 15.96262821942657382980226384649, 17.030790547738224515506380015419, 17.49152722181443056967305318920, 18.63499126115887267030550513939, 19.121924048187648472886159956140

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