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Properties

Label 1-4729-4729.1023-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.419 + 0.907i$
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  + (0.803 + 0.595i)2-s + (0.721 − 0.692i)3-s + (0.290 + 0.956i)4-s + (−0.0239 + 0.999i)5-s + (0.991 − 0.127i)6-s + (0.927 + 0.373i)7-s + (−0.336 + 0.941i)8-s + (0.0398 − 0.999i)9-s + (−0.614 + 0.788i)10-s + (−0.509 + 0.860i)11-s + (0.872 + 0.488i)12-s + (0.351 − 0.936i)13-s + (0.522 + 0.852i)14-s + (0.675 + 0.737i)15-s + (−0.830 + 0.556i)16-s + (0.783 − 0.620i)17-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (0.721 − 0.692i)3-s + (0.290 + 0.956i)4-s + (−0.0239 + 0.999i)5-s + (0.991 − 0.127i)6-s + (0.927 + 0.373i)7-s + (−0.336 + 0.941i)8-s + (0.0398 − 0.999i)9-s + (−0.614 + 0.788i)10-s + (−0.509 + 0.860i)11-s + (0.872 + 0.488i)12-s + (0.351 − 0.936i)13-s + (0.522 + 0.852i)14-s + (0.675 + 0.737i)15-s + (−0.830 + 0.556i)16-s + (0.783 − 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.190068553 + 3.424639868i\)
\(L(\frac12)\) \(\approx\) \(2.190068553 + 3.424639868i\)
\(L(1)\) \(\approx\) \(1.942864451 + 1.058989214i\)
\(L(1)\) \(\approx\) \(1.942864451 + 1.058989214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.803 + 0.595i)T \)
3 \( 1 + (0.721 - 0.692i)T \)
5 \( 1 + (-0.0239 + 0.999i)T \)
7 \( 1 + (0.927 + 0.373i)T \)
11 \( 1 + (-0.509 + 0.860i)T \)
13 \( 1 + (0.351 - 0.936i)T \)
17 \( 1 + (0.783 - 0.620i)T \)
19 \( 1 + (0.887 + 0.460i)T \)
23 \( 1 + (-0.709 + 0.704i)T \)
29 \( 1 + (0.803 - 0.595i)T \)
31 \( 1 + (-0.996 - 0.0796i)T \)
37 \( 1 + (-0.589 + 0.808i)T \)
41 \( 1 + (-0.978 - 0.205i)T \)
43 \( 1 + (-0.366 + 0.930i)T \)
47 \( 1 + (0.380 + 0.924i)T \)
53 \( 1 + (-0.880 + 0.474i)T \)
59 \( 1 + (0.821 + 0.569i)T \)
61 \( 1 + (0.763 + 0.645i)T \)
67 \( 1 + (-0.119 + 0.992i)T \)
71 \( 1 + (0.999 - 0.0318i)T \)
73 \( 1 + (-0.563 + 0.826i)T \)
79 \( 1 + T \)
83 \( 1 + (0.601 + 0.798i)T \)
89 \( 1 + (-0.536 - 0.843i)T \)
97 \( 1 + (0.927 + 0.373i)T \)
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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

−18.1633133492706323621455426246, −17.03794843549819876516875762340, −16.21306619016298524599043961848, −16.07479305001087222287260201903, −15.10957832117632209660833066375, −14.37529730291257608206621033937, −13.85373756277650402614306186093, −13.52439967675820274854121146585, −12.57648736628237849710148367996, −11.89157002209924926545974607270, −11.162276850728175677704078446036, −10.57287147789014781626820046706, −9.91054203156509748583978048679, −9.05584156006422098667981711309, −8.5013142398751041005545056833, −7.86380910327760308712295195003, −6.85273412514135756739136096010, −5.6854783484193569437842208425, −5.104425823683237595717816897946, −4.68446066952058632151723731439, −3.63513494888778821120278562174, −3.54428232011599121712528940915, −2.134001798303056582479771975942, −1.70847235105888911892165138378, −0.67762570551507361083703308988, 1.36311849774511987419301348557, 2.20460697120821499913044578543, 2.9137611052738081741931543542, 3.41290737162739171918537031661, 4.328613659928973904123431307963, 5.40537175299923242283828242051, 5.768806783025490769549546385100, 6.80816818986463230974271646102, 7.376631224283338028549744823392, 7.96931153054140790238763667551, 8.2319766973399313580819362480, 9.48933337710518650203674761124, 10.17809860359875141604230082474, 11.20362153416755013985138871782, 11.94086317997244559599687919774, 12.26984971565592148604587173876, 13.281127826348746178607293509419, 13.766561881106494398206253194431, 14.472119245649044635234492536271, 14.78473494228256148154564588345, 15.55830345306382444797639908720, 15.89498889988950226646945363560, 17.27254185050714503610265002825, 17.85685425279901557012782006717, 18.186806400756400784642456179650

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