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Properties

Label 1-4729-4729.1036-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.540 - 0.841i$
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  + (0.709 + 0.704i)2-s + (−0.949 − 0.313i)3-s + (0.00797 + 0.999i)4-s + (0.140 + 0.990i)5-s + (−0.453 − 0.891i)6-s + (0.631 + 0.775i)7-s + (−0.698 + 0.715i)8-s + (0.803 + 0.595i)9-s + (−0.597 + 0.801i)10-s + (0.768 + 0.639i)11-s + (0.305 − 0.952i)12-s + (−0.875 − 0.483i)13-s + (−0.0981 + 0.995i)14-s + (0.177 − 0.984i)15-s + (−0.999 + 0.0159i)16-s + (0.242 + 0.970i)17-s + ⋯
L(s)  = 1  + (0.709 + 0.704i)2-s + (−0.949 − 0.313i)3-s + (0.00797 + 0.999i)4-s + (0.140 + 0.990i)5-s + (−0.453 − 0.891i)6-s + (0.631 + 0.775i)7-s + (−0.698 + 0.715i)8-s + (0.803 + 0.595i)9-s + (−0.597 + 0.801i)10-s + (0.768 + 0.639i)11-s + (0.305 − 0.952i)12-s + (−0.875 − 0.483i)13-s + (−0.0981 + 0.995i)14-s + (0.177 − 0.984i)15-s + (−0.999 + 0.0159i)16-s + (0.242 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1036, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6354782078 + 1.164000206i\)
\(L(\frac12)\) \(\approx\) \(-0.6354782078 + 1.164000206i\)
\(L(1)\) \(\approx\) \(0.7623311360 + 0.8398107802i\)
\(L(1)\) \(\approx\) \(0.7623311360 + 0.8398107802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.709 + 0.704i)T \)
3 \( 1 + (-0.949 - 0.313i)T \)
5 \( 1 + (0.140 + 0.990i)T \)
7 \( 1 + (0.631 + 0.775i)T \)
11 \( 1 + (0.768 + 0.639i)T \)
13 \( 1 + (-0.875 - 0.483i)T \)
17 \( 1 + (0.242 + 0.970i)T \)
19 \( 1 + (0.749 - 0.661i)T \)
23 \( 1 + (-0.0637 + 0.997i)T \)
29 \( 1 + (-0.704 - 0.709i)T \)
31 \( 1 + (0.956 - 0.290i)T \)
37 \( 1 + (-0.129 + 0.991i)T \)
41 \( 1 + (0.174 - 0.984i)T \)
43 \( 1 + (-0.972 - 0.234i)T \)
47 \( 1 + (0.881 - 0.472i)T \)
53 \( 1 + (-0.450 + 0.892i)T \)
59 \( 1 + (-0.713 + 0.700i)T \)
61 \( 1 + (-0.685 + 0.728i)T \)
67 \( 1 + (-0.647 - 0.762i)T \)
71 \( 1 + (0.0132 - 0.999i)T \)
73 \( 1 + (-0.370 - 0.928i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.375 - 0.926i)T \)
89 \( 1 + (-0.987 + 0.156i)T \)
97 \( 1 + (-0.631 - 0.775i)T \)
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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

−17.67410988547640522654713066309, −16.89730032468917199354199385877, −16.42377058231898290425587615795, −15.94243116771142902650669064465, −14.80177748312658213539164527804, −14.209252936323112123254514474784, −13.73426670892054171386230529093, −12.77126240193625273017533999045, −12.23462339820773742519988607005, −11.65323739736837849550791333682, −11.201616409473186538982022183, −10.36641547111430889299146858860, −9.652017211409536773095795923939, −9.25147443943877342117121439361, −8.14750720691657749986300646466, −7.10484433937828798782416713055, −6.46679449720705400379981889214, −5.51039552766609448883297201904, −5.08703818385370654333498610883, −4.393679853739126585436584474596, −3.96050601543604621438799261659, −2.92976835321432201679966906921, −1.620833931418219983385442478164, −1.17420349780905446851331756926, −0.31721934354165124204778947743, 1.52518846005755041506162427586, 2.27668833180681335880955951524, 3.11996258013803350331912114771, 4.10679121625674401296785237563, 4.84147057016961026874923498617, 5.550482528549180787045186714121, 6.07209084802445589160245270754, 6.73259795846136359445604793567, 7.60756841734879893303719492041, 7.71439972194042586168626537899, 9.0136970104477367095319385073, 9.8188744871852133399053009238, 10.64496978224513914421600314392, 11.48532606241591183272053481821, 12.00345758887371008104182378565, 12.294450083267708204520924416114, 13.42318464330508339663797481213, 13.830944313505608489922842111951, 14.82430136133392842399370258876, 15.319220709939558329674405484594, 15.49408684159581564896186522213, 16.903148543532777308805201611928, 17.187917092061963060854951787439, 17.753355904282104530428475310750, 18.28144381563708663102662888552

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