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Properties

Label 1-4729-4729.10-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.995 - 0.0958i$
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  + (−0.773 − 0.633i)2-s + (−0.119 + 0.992i)3-s + (0.198 + 0.980i)4-s + (0.618 − 0.785i)5-s + (0.721 − 0.692i)6-s + (−0.316 − 0.948i)7-s + (0.467 − 0.884i)8-s + (−0.971 − 0.236i)9-s + (−0.976 + 0.216i)10-s + (−0.997 − 0.0637i)11-s + (−0.996 + 0.0796i)12-s + (0.448 + 0.893i)13-s + (−0.356 + 0.934i)14-s + (0.706 + 0.708i)15-s + (−0.921 + 0.388i)16-s + (0.346 + 0.938i)17-s + ⋯
L(s)  = 1  + (−0.773 − 0.633i)2-s + (−0.119 + 0.992i)3-s + (0.198 + 0.980i)4-s + (0.618 − 0.785i)5-s + (0.721 − 0.692i)6-s + (−0.316 − 0.948i)7-s + (0.467 − 0.884i)8-s + (−0.971 − 0.236i)9-s + (−0.976 + 0.216i)10-s + (−0.997 − 0.0637i)11-s + (−0.996 + 0.0796i)12-s + (0.448 + 0.893i)13-s + (−0.356 + 0.934i)14-s + (0.706 + 0.708i)15-s + (−0.921 + 0.388i)16-s + (0.346 + 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.995 - 0.0958i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.995 - 0.0958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02032991833 - 0.4233581772i\)
\(L(\frac12)\) \(\approx\) \(0.02032991833 - 0.4233581772i\)
\(L(1)\) \(\approx\) \(0.6167320697 - 0.1664155694i\)
\(L(1)\) \(\approx\) \(0.6167320697 - 0.1664155694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.773 - 0.633i)T \)
3 \( 1 + (-0.119 + 0.992i)T \)
5 \( 1 + (0.618 - 0.785i)T \)
7 \( 1 + (-0.316 - 0.948i)T \)
11 \( 1 + (-0.997 - 0.0637i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.346 + 0.938i)T \)
19 \( 1 + (0.249 - 0.968i)T \)
23 \( 1 + (0.0239 - 0.999i)T \)
29 \( 1 + (0.773 - 0.633i)T \)
31 \( 1 + (-0.887 - 0.460i)T \)
37 \( 1 + (-0.917 + 0.397i)T \)
41 \( 1 + (-0.321 - 0.947i)T \)
43 \( 1 + (0.988 + 0.153i)T \)
47 \( 1 + (0.968 - 0.247i)T \)
53 \( 1 + (-0.341 - 0.939i)T \)
59 \( 1 + (0.0292 + 0.999i)T \)
61 \( 1 + (0.518 - 0.855i)T \)
67 \( 1 + (-0.192 + 0.981i)T \)
71 \( 1 + (-0.655 - 0.755i)T \)
73 \( 1 + (-0.824 + 0.565i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.951 - 0.308i)T \)
89 \( 1 + (0.265 - 0.964i)T \)
97 \( 1 + (-0.316 - 0.948i)T \)
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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

−18.374540701443359226108273757962, −17.901650361195884710875122844534, −17.55485878024516689115266960797, −16.47035037860152626602436282832, −15.857148260239988804676055405363, −15.26545349646476742102051061002, −14.423174213047959861861621042971, −13.93383128009841624467327271042, −13.21003899941319831645177197154, −12.453175434308139692147401769356, −11.69100458698761827476899586801, −10.821317412938582543238198239404, −10.36547154747519307187628762818, −9.50102356964418349690719287712, −8.86670168660001975670163741206, −8.007308172552377496976865365296, −7.5020276249676394149347288311, −6.87896923314810631927625051856, −6.016135176244552952599581763554, −5.56422330064353153847334388878, −5.206499337863990510898705459124, −3.197108863304495183558113321828, −2.76471792078102888809237440918, −1.90707034945186861418036866178, −1.17194022285796894761952464603, 0.16946246900569464897982929603, 1.03133246956080162767138915898, 2.09926337804085334092465419128, 2.84333878246343450738912607934, 3.8498507904867895486816246376, 4.29090302203508474083167334379, 5.084105123308181286982808926982, 6.02670385009382509594992007672, 6.82648178673050771720297701069, 7.775499275912069179186209798081, 8.65600606023155430070953866648, 8.94101109642635341381770433435, 9.84567759087697996668529755442, 10.308766917371623447309084756554, 10.7587947923919970613439643176, 11.53001775297054845332340230592, 12.3773002813670587365417920292, 13.08312778250690797473280294390, 13.65626858196291101074981860447, 14.35919178687820293719936828273, 15.59083575081550781279853654953, 16.05564827686502028529065415585, 16.579607249816546255357377356082, 17.2661254752106411657873973234, 17.51133463642385026803495609893

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