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Properties

Label 1-4729-4729.295-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.973 + 0.230i$
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  + (−0.395 + 0.918i)2-s + (0.439 − 0.898i)3-s + (−0.687 − 0.726i)4-s + (0.229 − 0.973i)5-s + (0.651 + 0.758i)6-s + (−0.848 − 0.529i)7-s + (0.939 − 0.343i)8-s + (−0.614 − 0.788i)9-s + (0.803 + 0.595i)10-s + (0.0717 + 0.997i)11-s + (−0.954 + 0.298i)12-s + (0.321 − 0.947i)13-s + (0.821 − 0.569i)14-s + (−0.773 − 0.633i)15-s + (−0.0557 + 0.998i)16-s + (0.981 − 0.190i)17-s + ⋯
L(s)  = 1  + (−0.395 + 0.918i)2-s + (0.439 − 0.898i)3-s + (−0.687 − 0.726i)4-s + (0.229 − 0.973i)5-s + (0.651 + 0.758i)6-s + (−0.848 − 0.529i)7-s + (0.939 − 0.343i)8-s + (−0.614 − 0.788i)9-s + (0.803 + 0.595i)10-s + (0.0717 + 0.997i)11-s + (−0.954 + 0.298i)12-s + (0.321 − 0.947i)13-s + (0.821 − 0.569i)14-s + (−0.773 − 0.633i)15-s + (−0.0557 + 0.998i)16-s + (0.981 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07672719158 - 0.6579284932i\)
\(L(\frac12)\) \(\approx\) \(-0.07672719158 - 0.6579284932i\)
\(L(1)\) \(\approx\) \(0.7624622012 - 0.2538604828i\)
\(L(1)\) \(\approx\) \(0.7624622012 - 0.2538604828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.395 + 0.918i)T \)
3 \( 1 + (0.439 - 0.898i)T \)
5 \( 1 + (0.229 - 0.973i)T \)
7 \( 1 + (-0.848 - 0.529i)T \)
11 \( 1 + (0.0717 + 0.997i)T \)
13 \( 1 + (0.321 - 0.947i)T \)
17 \( 1 + (0.981 - 0.190i)T \)
19 \( 1 + (-0.0875 - 0.996i)T \)
23 \( 1 + (0.975 - 0.221i)T \)
29 \( 1 + (-0.395 - 0.918i)T \)
31 \( 1 + (-0.244 + 0.969i)T \)
37 \( 1 + (-0.753 + 0.657i)T \)
41 \( 1 + (0.576 + 0.817i)T \)
43 \( 1 + (-0.536 - 0.843i)T \)
47 \( 1 + (-0.993 + 0.111i)T \)
53 \( 1 + (-0.830 + 0.556i)T \)
59 \( 1 + (-0.812 - 0.582i)T \)
61 \( 1 + (-0.0239 - 0.999i)T \)
67 \( 1 + (0.915 - 0.402i)T \)
71 \( 1 + (-0.213 + 0.976i)T \)
73 \( 1 + (-0.481 - 0.876i)T \)
79 \( 1 + T \)
83 \( 1 + (0.0398 - 0.999i)T \)
89 \( 1 + (0.959 + 0.283i)T \)
97 \( 1 + (-0.848 - 0.529i)T \)
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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

−18.82876139599405732329478509807, −18.15649894190111004377235819022, −17.07668278783785298173760859889, −16.43684838460262575360246630791, −16.15060497158618311413738204270, −14.99649617750342968064958642860, −14.454315798738247291961150049281, −13.83825089867868941137006385100, −13.20372830163313522840975074755, −12.31761600800781864661576220617, −11.49084872581171303769397700761, −10.95447638032030078621646942851, −10.404998911544646359833146918114, −9.600485667882743172721460074500, −9.29532222223292768341977019197, −8.52807715991350357134223665793, −7.79946209064505218629612300528, −6.856493418895348923884597017036, −5.878437203312932865306301635914, −5.31498143570800861324654020897, −4.04290663648004232636637175055, −3.46464055809819574389710570000, −3.136690313054173119152913207936, −2.30190931330168682067549224740, −1.45374342871652145022303938398, 0.21142871755197953902466595770, 1.01874560387551781901781296788, 1.66450133721056764106239631077, 2.885351013335273076188421941012, 3.71472260544696543967246623818, 4.80634830477149739057157201951, 5.31677916466751998214990869, 6.312846562043823307769711252016, 6.74139975068982329789674248494, 7.59392515359535420623667774435, 7.991829729314605578843477794518, 8.86368021357156869814082137102, 9.43200406371449584112175530393, 9.93923761948993648388791029874, 10.83415991750100897290489825830, 12.0500790649884749627282184281, 12.7914670244268530433231241267, 13.09199009327869574377215326539, 13.70773146603031868056007650181, 14.44185654200089856839593863557, 15.273341421787682422547087049468, 15.7418064148274903726505320941, 16.6265486293371023945268541154, 17.27318615834281862581048768807, 17.54596489499958625332225853608

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