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Properties

Label 1-539-539.114-r0-0-0
Degree $1$
Conductor $539$
Sign $0.907 - 0.420i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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 + (0.887 + 0.460i)3-s + (0.525 + 0.850i)4-s + (−0.772 − 0.635i)5-s + (−0.550 − 0.834i)6-s + (−0.0448 − 0.998i)8-s + (0.575 + 0.817i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (−0.447 + 0.894i)16-s + (−0.280 − 0.959i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + (0.134 − 0.990i)20-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)2-s + (0.887 + 0.460i)3-s + (0.525 + 0.850i)4-s + (−0.772 − 0.635i)5-s + (−0.550 − 0.834i)6-s + (−0.0448 − 0.998i)8-s + (0.575 + 0.817i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (−0.447 + 0.894i)16-s + (−0.280 − 0.959i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + (0.134 − 0.990i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.907 - 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.108857014 - 0.2442234778i\)
\(L(\frac12)\) \(\approx\) \(1.108857014 - 0.2442234778i\)
\(L(1)\) \(\approx\) \(0.9114944972 - 0.1269449523i\)
\(L(1)\) \(\approx\) \(0.9114944972 - 0.1269449523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.873 - 0.486i)T \)
3 \( 1 + (0.887 + 0.460i)T \)
5 \( 1 + (-0.772 - 0.635i)T \)
13 \( 1 + (0.858 - 0.512i)T \)
17 \( 1 + (-0.280 - 0.959i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.646 - 0.762i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.992 - 0.119i)T \)
53 \( 1 + (0.971 - 0.237i)T \)
59 \( 1 + (-0.842 - 0.538i)T \)
61 \( 1 + (0.971 + 0.237i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.992 + 0.119i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.858 + 0.512i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−23.74573227788629915563218978613, −23.07001477013750255564618899669, −21.66448040671202027476444346978, −20.64849713307570799939771627540, −19.699032022259350660255092762774, −19.32410050381721790624872225232, −18.45930433797863235578008289353, −17.90476513076420588891708816626, −16.70596002633630599810219780122, −15.66569219305991572284959491493, −15.13229487020497231199696217519, −14.36841043014311298022917876413, −13.44707043762926405321183727913, −12.24354309035295278978281665607, −11.13056380118295512382140202621, −10.47944412116879478389157897475, −9.15242240848718701303132300450, −8.57191633990309034486992888281, −7.75705875272483537489704670524, −6.8241678010709101616063417806, −6.33282517062941568310055285916, −4.557769864912268814465076587771, −3.32932643563186794201953192702, −2.30331794355416873473602751424, −1.0572643758375900708772765310, 0.94786499166891709208969762096, 2.23324005730816147742996217074, 3.44224889553934842974820580288, 3.996525535089576166365795718557, 5.34221940358437812180856535286, 7.119642655945440168099673721811, 7.8219211019966602832583996061, 8.72345640546277464360389242523, 9.154260992609850133095863988580, 10.314490321689586743558986710743, 11.06240791845011172627809302322, 12.1037798716881368278805607350, 12.9545081129065791189958565916, 13.87581190794556731509589266736, 15.22057813732580806816943207153, 15.87837196735496056854155093564, 16.43176105521813599818509836670, 17.527925916289829689643045767640, 18.6573162556923181732595035046, 19.21437957655525042500051617554, 20.13517959081797200188161623235, 20.624031657476517152187918284590, 21.202774573592990532139003581932, 22.35625965443450216113545321152, 23.34501682280426873987533075939

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