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Properties

Label 1-4729-4729.19-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.792 - 0.610i$
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  + (−0.864 + 0.502i)2-s + (−0.305 − 0.952i)3-s + (0.495 − 0.868i)4-s + (−0.702 + 0.711i)5-s + (0.742 + 0.669i)6-s + (0.994 − 0.106i)7-s + (0.00797 + 0.999i)8-s + (−0.812 + 0.582i)9-s + (0.249 − 0.968i)10-s + (−0.894 − 0.446i)11-s + (−0.978 − 0.205i)12-s + (0.935 − 0.353i)13-s + (−0.806 + 0.591i)14-s + (0.892 + 0.450i)15-s + (−0.509 − 0.860i)16-s + (0.331 + 0.943i)17-s + ⋯
L(s)  = 1  + (−0.864 + 0.502i)2-s + (−0.305 − 0.952i)3-s + (0.495 − 0.868i)4-s + (−0.702 + 0.711i)5-s + (0.742 + 0.669i)6-s + (0.994 − 0.106i)7-s + (0.00797 + 0.999i)8-s + (−0.812 + 0.582i)9-s + (0.249 − 0.968i)10-s + (−0.894 − 0.446i)11-s + (−0.978 − 0.205i)12-s + (0.935 − 0.353i)13-s + (−0.806 + 0.591i)14-s + (0.892 + 0.450i)15-s + (−0.509 − 0.860i)16-s + (0.331 + 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.792 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7536570439 - 0.2566253027i\)
\(L(\frac12)\) \(\approx\) \(0.7536570439 - 0.2566253027i\)
\(L(1)\) \(\approx\) \(0.6220743016 + 0.02272888797i\)
\(L(1)\) \(\approx\) \(0.6220743016 + 0.02272888797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.864 + 0.502i)T \)
3 \( 1 + (-0.305 - 0.952i)T \)
5 \( 1 + (-0.702 + 0.711i)T \)
7 \( 1 + (0.994 - 0.106i)T \)
11 \( 1 + (-0.894 - 0.446i)T \)
13 \( 1 + (0.935 - 0.353i)T \)
17 \( 1 + (0.331 + 0.943i)T \)
19 \( 1 + (0.610 + 0.792i)T \)
23 \( 1 + (-0.536 + 0.843i)T \)
29 \( 1 + (-0.864 - 0.502i)T \)
31 \( 1 + (0.321 - 0.947i)T \)
37 \( 1 + (0.906 + 0.422i)T \)
41 \( 1 + (-0.395 - 0.918i)T \)
43 \( 1 + (-0.655 - 0.755i)T \)
47 \( 1 + (0.999 + 0.0212i)T \)
53 \( 1 + (-0.999 + 0.0372i)T \)
59 \( 1 + (-0.647 + 0.762i)T \)
61 \( 1 + (-0.988 - 0.153i)T \)
67 \( 1 + (0.683 - 0.730i)T \)
71 \( 1 + (0.942 - 0.333i)T \)
73 \( 1 + (-0.580 + 0.814i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.904 - 0.427i)T \)
89 \( 1 + (0.970 + 0.242i)T \)
97 \( 1 + (0.994 - 0.106i)T \)
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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

−18.34687184695889622389488922170, −17.589581964361839885000769036822, −16.86873265123743410776463987517, −16.146660546051620084159377282816, −15.89171051628626035607861724376, −15.21935271337936258040795608761, −14.34321952306314266833531317395, −13.393673708995926345698553848095, −12.51231941477152459301166276593, −11.89833088594763171289813619181, −11.19999984206507428438545422856, −11.00027410178775381347488658410, −10.06067295971627378121544726100, −9.31633600459177983642968406771, −8.80123102156801271554305138056, −8.09504136281603159745653090256, −7.59675526470877340299454282961, −6.61499213232660107630862265800, −5.46102386921955321770332161298, −4.77385819588043052185172646502, −4.2951864668540470033259873337, −3.35028079561691092280639933280, −2.64542199900841774761006904216, −1.50860986827171138006198825392, −0.69093924075147710087980296284, 0.46592499925801944639511897772, 1.43381743867240774055303105505, 2.06062862272460060229757956969, 3.0532311103734552563342144159, 3.99253347712584387206946378270, 5.25735737132060921141574608550, 5.86041452663588971147461618944, 6.30519266221266864015971259458, 7.491911878947639249757192774979, 7.75672416272092433205064241029, 8.10574907525295531806345250942, 8.87277590528645142967087880208, 10.197054716769077699350549167878, 10.61136315424096152210647268658, 11.40056756423521630830956549326, 11.60433143774023932906464895096, 12.60615107860475453507638721395, 13.72419649179853267455780428432, 13.96153237561674326825200610237, 14.98732042321250985243425027981, 15.384514106471866797706239751397, 16.1536044766112235334263313446, 16.99110771571844850862227839259, 17.45912043651286370379146158067, 18.332859958601134762192423705198

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