Properties

Label 1-837-837.727-r0-0-0
Degree $1$
Conductor $837$
Sign $0.522 - 0.852i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.559 + 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.939 + 0.342i)5-s + (−0.882 + 0.469i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.848 + 0.529i)11-s + (−0.997 − 0.0697i)13-s + (−0.882 − 0.469i)14-s + (−0.719 − 0.694i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.0348 − 0.999i)20-s + (0.0348 + 0.999i)22-s + (0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (0.559 + 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.939 + 0.342i)5-s + (−0.882 + 0.469i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.848 + 0.529i)11-s + (−0.997 − 0.0697i)13-s + (−0.882 − 0.469i)14-s + (−0.719 − 0.694i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (0.0348 − 0.999i)20-s + (0.0348 + 0.999i)22-s + (0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09052162189 - 0.05071434356i\)
\(L(\frac12)\) \(\approx\) \(0.09052162189 - 0.05071434356i\)
\(L(1)\) \(\approx\) \(0.6358607692 + 0.4540727649i\)
\(L(1)\) \(\approx\) \(0.6358607692 + 0.4540727649i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.559 + 0.829i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.882 + 0.469i)T \)
11 \( 1 + (0.848 + 0.529i)T \)
13 \( 1 + (-0.997 - 0.0697i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + (0.559 + 0.829i)T \)
47 \( 1 + (-0.241 + 0.970i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.997 - 0.0697i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.374 - 0.927i)T \)
83 \( 1 + (0.438 - 0.898i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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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

−22.29567844719641662211266210612, −21.5727536352501128809267739136, −20.51424586303718816209851553655, −19.83469273137685117195000837418, −19.29458366047894567104336355242, −18.84750992517736074308346238409, −17.35709380270108398531089189202, −16.68959145888541851908532825057, −15.62654714376701946546624071719, −14.99781420789664592077644760015, −14.00852746626440013669125186488, −13.19922862527259113799839892837, −12.4574257524021215114476101662, −11.734914788887697385199166074923, −11.02126540715285697662356861734, −10.02940900703073334042171727649, −9.233774844860303242218803538008, −8.37191941069251113132825943166, −7.01289360030174393899095582402, −6.3434737541186246580811437354, −5.03951141310332274065071589359, −4.19611840556414579042203171512, −3.546297458841793257073469592241, −2.567331155298338162053495434326, −1.15464235845553914440556495973, 0.04219650045041716881365930878, 2.42010295156675898278965184765, 3.26625696261632147775813424394, 4.28160331283068429516282613116, 4.91517964479406220819464187692, 6.35584048035190313433494832223, 6.801146541030231282316701541010, 7.563726992846697181797162946156, 8.775428662083665464394362483002, 9.23494404510957028772914705794, 10.62787750588607384277275760818, 11.6610939384546656407285875556, 12.56211806178466029159694184863, 12.8203854096153113889231326159, 14.24754464960501016573324149298, 14.83615612408903991615487655075, 15.53148757767008685434238633478, 16.13153252566435385106351052676, 17.13213670590108325503764173709, 17.736160361428059128527607557855, 18.96994825880279226368957485577, 19.51091625630147207841047342284, 20.35339725605381193502861646622, 21.677055141974897057075821763843, 22.229998177450852714688037678454

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