Properties

Label 1-4729-4729.73-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.966 - 0.256i$
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  + (−0.991 − 0.127i)2-s + (0.709 − 0.704i)3-s + (0.967 + 0.252i)4-s + (0.746 − 0.665i)5-s + (−0.793 + 0.608i)6-s + (−0.610 − 0.792i)7-s + (−0.927 − 0.373i)8-s + (0.00797 − 0.999i)9-s + (−0.824 + 0.565i)10-s + (0.205 + 0.978i)11-s + (0.864 − 0.502i)12-s + (0.827 + 0.560i)13-s + (0.504 + 0.863i)14-s + (0.0610 − 0.998i)15-s + (0.872 + 0.488i)16-s + (0.525 + 0.851i)17-s + ⋯
L(s)  = 1  + (−0.991 − 0.127i)2-s + (0.709 − 0.704i)3-s + (0.967 + 0.252i)4-s + (0.746 − 0.665i)5-s + (−0.793 + 0.608i)6-s + (−0.610 − 0.792i)7-s + (−0.927 − 0.373i)8-s + (0.00797 − 0.999i)9-s + (−0.824 + 0.565i)10-s + (0.205 + 0.978i)11-s + (0.864 − 0.502i)12-s + (0.827 + 0.560i)13-s + (0.504 + 0.863i)14-s + (0.0610 − 0.998i)15-s + (0.872 + 0.488i)16-s + (0.525 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.634733113 - 0.2131134545i\)
\(L(\frac12)\) \(\approx\) \(1.634733113 - 0.2131134545i\)
\(L(1)\) \(\approx\) \(0.9799025509 - 0.2818028701i\)
\(L(1)\) \(\approx\) \(0.9799025509 - 0.2818028701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.991 - 0.127i)T \)
3 \( 1 + (0.709 - 0.704i)T \)
5 \( 1 + (0.746 - 0.665i)T \)
7 \( 1 + (-0.610 - 0.792i)T \)
11 \( 1 + (0.205 + 0.978i)T \)
13 \( 1 + (0.827 + 0.560i)T \)
17 \( 1 + (0.525 + 0.851i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (0.127 + 0.991i)T \)
31 \( 1 + (-0.0159 + 0.999i)T \)
37 \( 1 + (0.999 + 0.0212i)T \)
41 \( 1 + (-0.620 + 0.783i)T \)
43 \( 1 + (-0.722 + 0.690i)T \)
47 \( 1 + (-0.879 + 0.476i)T \)
53 \( 1 + (0.303 + 0.952i)T \)
59 \( 1 + (0.224 - 0.974i)T \)
61 \( 1 + (-0.0690 - 0.997i)T \)
67 \( 1 + (0.877 - 0.479i)T \)
71 \( 1 + (0.0981 - 0.995i)T \)
73 \( 1 + (0.800 + 0.599i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.285 + 0.958i)T \)
89 \( 1 + (-0.216 - 0.976i)T \)
97 \( 1 + (0.610 + 0.792i)T \)
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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

−18.44208319703883783020659042265, −17.51822378510080066704749161076, −16.77364128357288953193939478719, −16.235167565195641748195411484805, −15.42580397467864476719320936478, −15.143319557286569094677209706467, −14.39838983710502943362073817482, −13.39746654058414299216842691288, −13.19223138354861678927943266907, −11.63955476391794878621122285113, −11.29242072060997929905751833194, −10.46755768529029248175421984308, −9.88380655800749290701097473227, −9.32592560658777455094696364744, −8.739132080568194867225487164937, −8.20908842966147239075326814318, −7.22555253447738679462331149298, −6.51014879930262609099342930859, −5.716293529670123539034476997123, −5.32069299467426358761540420436, −3.78591551633146586061678683918, −2.98222601376750626261322152565, −2.69695837456390960005458726560, −1.82684299034829493770944255323, −0.584070282907253751635733932973, 1.08353359285270446844977048195, 1.41097616008019673079044332665, 2.110090033004644967328263547063, 3.15177947493304931665466593438, 3.776904356749944181992693935366, 4.80311469091531036171264288488, 6.22196582378698298635344747913, 6.45212731977383740971328245897, 7.142010668077986963023482948763, 8.082128748226032113191156188335, 8.49990252870610883526729822392, 9.30634310384333326267807665137, 9.776234796184671926806899236112, 10.403148806747110656505371681644, 11.23359419332082736790676726115, 12.38960921280374658272468296572, 12.66236601333618877901760193117, 13.18348810405100214534473029239, 14.214904719891719804607301626807, 14.64652731151178938409109496632, 15.58061505867850490648254797889, 16.49469767095585425468664583887, 16.86539104440386954908920487167, 17.467545867538750448016860040875, 18.2826621798827853002943780201

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