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Properties

Label 1-4729-4729.1052-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.999 + 0.00168i$
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  + (−0.0557 − 0.998i)2-s + (−0.244 − 0.969i)3-s + (−0.993 + 0.111i)4-s + (0.390 + 0.920i)5-s + (−0.954 + 0.298i)6-s + (0.990 − 0.137i)7-s + (0.166 + 0.986i)8-s + (−0.880 + 0.474i)9-s + (0.897 − 0.441i)10-s + (0.959 + 0.283i)11-s + (0.351 + 0.936i)12-s + (0.706 + 0.708i)13-s + (−0.192 − 0.981i)14-s + (0.796 − 0.604i)15-s + (0.975 − 0.221i)16-s + (−0.960 + 0.278i)17-s + ⋯
L(s)  = 1  + (−0.0557 − 0.998i)2-s + (−0.244 − 0.969i)3-s + (−0.993 + 0.111i)4-s + (0.390 + 0.920i)5-s + (−0.954 + 0.298i)6-s + (0.990 − 0.137i)7-s + (0.166 + 0.986i)8-s + (−0.880 + 0.474i)9-s + (0.897 − 0.441i)10-s + (0.959 + 0.283i)11-s + (0.351 + 0.936i)12-s + (0.706 + 0.708i)13-s + (−0.192 − 0.981i)14-s + (0.796 − 0.604i)15-s + (0.975 − 0.221i)16-s + (−0.960 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.999 + 0.00168i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.999 + 0.00168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.578702335 + 0.001326346016i\)
\(L(\frac12)\) \(\approx\) \(1.578702335 + 0.001326346016i\)
\(L(1)\) \(\approx\) \(0.9713222015 - 0.3944872883i\)
\(L(1)\) \(\approx\) \(0.9713222015 - 0.3944872883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.0557 - 0.998i)T \)
3 \( 1 + (-0.244 - 0.969i)T \)
5 \( 1 + (0.390 + 0.920i)T \)
7 \( 1 + (0.990 - 0.137i)T \)
11 \( 1 + (0.959 + 0.283i)T \)
13 \( 1 + (0.706 + 0.708i)T \)
17 \( 1 + (-0.960 + 0.278i)T \)
19 \( 1 + (-0.767 + 0.641i)T \)
23 \( 1 + (0.627 + 0.778i)T \)
29 \( 1 + (-0.0557 + 0.998i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
37 \( 1 + (0.249 - 0.968i)T \)
41 \( 1 + (-0.773 + 0.633i)T \)
43 \( 1 + (0.985 + 0.169i)T \)
47 \( 1 + (-0.824 + 0.565i)T \)
53 \( 1 + (0.964 + 0.262i)T \)
59 \( 1 + (-0.129 + 0.991i)T \)
61 \( 1 + (-0.414 - 0.909i)T \)
67 \( 1 + (0.906 - 0.422i)T \)
71 \( 1 + (-0.982 - 0.184i)T \)
73 \( 1 + (-0.571 - 0.820i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.631 - 0.775i)T \)
89 \( 1 + (0.584 + 0.811i)T \)
97 \( 1 + (0.990 - 0.137i)T \)
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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

−17.67144303384203964247196621549, −17.337235774611813502733608528888, −16.92578865802003419975702116785, −16.11209267705277041900462066481, −15.522033630258963552631149687665, −15.023200550512792077351922267759, −14.30006976894431747080757508189, −13.6313490125553770992773579905, −13.018779956275445806943078732803, −12.045383998940051959677386535648, −11.341031635599064908481546114488, −10.57340366704511935680886114474, −9.840596767122197987014508732309, −8.98375901680611967546004395764, −8.53968395701076512466582174728, −8.33417997705219305295402001224, −6.93821031443154665433181020601, −6.25125377222404423170870916417, −5.59146439525276048361283120115, −4.88746264183980465189502287229, −4.4351778707672733479942830603, −3.82694637419119559316057654910, −2.583341259013519192180683916978, −1.29097228732621909604685810727, −0.50111941738993570548963136388, 1.12912647684294189181343591408, 1.7634342593134737477863425318, 2.13181348282901837349327724216, 3.162655512089142384451821009138, 4.02428724825205562523681288189, 4.70666400537605461709428332874, 5.79656775489282240249112600593, 6.34612751950128162619691302824, 7.17803343340031214761524949463, 7.85662106524371961971284307072, 8.77397665000745548593320933178, 9.17687260104107624240755175749, 10.30785604769434153098786856352, 10.97728379928407970755631203881, 11.360084496026500035338878192229, 11.86166685520811203032925758724, 12.7661923872319956886370615189, 13.431300903080022808315291225427, 13.97475706557245586657408000293, 14.57026429931558285844672180426, 15.0402909476289138970298498016, 16.54124980452718283797580176837, 17.29597177925271272739660719536, 17.62406514987083124662270346554, 18.2468428508590580165824659868

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