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Properties

Label 1-4729-4729.3282-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: $\chi_{4729} (3282, \cdot )$
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

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

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