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Properties

Label 1-4729-4729.14-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.762 + 0.646i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.880 + 0.474i)2-s + (−0.275 + 0.961i)3-s + (0.549 − 0.835i)4-s + (0.756 − 0.653i)5-s + (−0.213 − 0.976i)6-s + (0.390 + 0.920i)7-s + (−0.0875 + 0.996i)8-s + (−0.848 − 0.529i)9-s + (−0.356 + 0.934i)10-s + (−0.366 − 0.930i)11-s + (0.651 + 0.758i)12-s + (−0.671 − 0.741i)13-s + (−0.780 − 0.624i)14-s + (0.419 + 0.907i)15-s + (−0.395 − 0.918i)16-s + (0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (−0.880 + 0.474i)2-s + (−0.275 + 0.961i)3-s + (0.549 − 0.835i)4-s + (0.756 − 0.653i)5-s + (−0.213 − 0.976i)6-s + (0.390 + 0.920i)7-s + (−0.0875 + 0.996i)8-s + (−0.848 − 0.529i)9-s + (−0.356 + 0.934i)10-s + (−0.366 − 0.930i)11-s + (0.651 + 0.758i)12-s + (−0.671 − 0.741i)13-s + (−0.780 − 0.624i)14-s + (0.419 + 0.907i)15-s + (−0.395 − 0.918i)16-s + (0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.762 + 0.646i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9063681284 + 0.3326306411i\)
\(L(\frac12)\) \(\approx\) \(0.9063681284 + 0.3326306411i\)
\(L(1)\) \(\approx\) \(0.6675293864 + 0.2363308312i\)
\(L(1)\) \(\approx\) \(0.6675293864 + 0.2363308312i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.880 + 0.474i)T \)
3 \( 1 + (-0.275 + 0.961i)T \)
5 \( 1 + (0.756 - 0.653i)T \)
7 \( 1 + (0.390 + 0.920i)T \)
11 \( 1 + (-0.366 - 0.930i)T \)
13 \( 1 + (-0.671 - 0.741i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.806 + 0.591i)T \)
23 \( 1 + (-0.0557 + 0.998i)T \)
29 \( 1 + (-0.880 - 0.474i)T \)
31 \( 1 + (0.439 + 0.898i)T \)
37 \( 1 + (0.0823 + 0.996i)T \)
41 \( 1 + (-0.971 + 0.236i)T \)
43 \( 1 + (0.999 + 0.0106i)T \)
47 \( 1 + (0.972 - 0.231i)T \)
53 \( 1 + (-0.917 - 0.397i)T \)
59 \( 1 + (0.994 + 0.106i)T \)
61 \( 1 + (0.124 + 0.992i)T \)
67 \( 1 + (-0.913 + 0.407i)T \)
71 \( 1 + (-0.824 - 0.565i)T \)
73 \( 1 + (0.504 + 0.863i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.140 - 0.990i)T \)
89 \( 1 + (-0.899 - 0.436i)T \)
97 \( 1 + (0.390 + 0.920i)T \)
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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

−18.16104340241026161794553819590, −17.34608844996418356924172241930, −17.1149464888467669016258266116, −16.664137348868862026742326815521, −15.38821049510117359780638962219, −14.54298074589862585240636814864, −14.083499009927995233646823100254, −13.039514303699469319037732590124, −12.76706550955119632161066705633, −11.96587928982857953354964846433, −11.07099765271153077615259890874, −10.69554926990690014606141506351, −10.062278501710200191812221136814, −9.311332981185432654400464253440, −8.450522096379429488304845961503, −7.580983104720295353146403306782, −7.21691798850945720564673749574, −6.62883632811855853746814137356, −5.893846584276894589631179534630, −4.74419384769527007142684138052, −3.919183404874716618099415827305, −2.77104794000700781706437641387, −2.042774350235776162791069956658, −1.77009670081852891487557630820, −0.62851801437404331183383896168, 0.5475224862136182045678124066, 1.58409314813676281058165334178, 2.54809988539388361229152602557, 3.20244842607206352676838541979, 4.63748257139825927735323113707, 5.23204283767482161422939830693, 5.761374496111839673169102457, 6.103719431109761117138696199342, 7.404706236222723032556136660212, 8.2591883023681726264309087547, 8.723477587105313378925146272084, 9.3185820200582568960995416329, 10.087870926639535332429084821652, 10.365673207201249807319149984144, 11.45589959048925699038237088947, 11.81997079880153974214085513123, 12.7774781024882655121950788603, 13.81374533629720165148760749054, 14.42998553201863797457984917344, 15.151691693771482311537671737008, 15.67392619829827656591632944547, 16.34285711811995650277179223977, 16.86750309395017754023588456863, 17.50386766699220375938857848503, 17.962884353064661037180744576373

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