Properties

Label 1-4729-4729.4257-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: $\chi_{4729} (4257, \cdot )$
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

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

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