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Properties

Label 1-4729-4729.2686-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.846 + 0.531i$
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  + (0.275 + 0.961i)2-s + (0.944 + 0.328i)3-s + (−0.848 + 0.529i)4-s + (0.0875 − 0.996i)5-s + (−0.0557 + 0.998i)6-s + (−0.166 − 0.986i)7-s + (−0.742 − 0.669i)8-s + (0.783 + 0.620i)9-s + (0.981 − 0.190i)10-s + (0.990 − 0.135i)11-s + (−0.975 + 0.221i)12-s + (0.967 − 0.252i)13-s + (0.901 − 0.431i)14-s + (0.410 − 0.912i)15-s + (0.439 − 0.898i)16-s + (0.633 − 0.773i)17-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (0.944 + 0.328i)3-s + (−0.848 + 0.529i)4-s + (0.0875 − 0.996i)5-s + (−0.0557 + 0.998i)6-s + (−0.166 − 0.986i)7-s + (−0.742 − 0.669i)8-s + (0.783 + 0.620i)9-s + (0.981 − 0.190i)10-s + (0.990 − 0.135i)11-s + (−0.975 + 0.221i)12-s + (0.967 − 0.252i)13-s + (0.901 − 0.431i)14-s + (0.410 − 0.912i)15-s + (0.439 − 0.898i)16-s + (0.633 − 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.846 + 0.531i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2686, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.846 + 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.206055418 + 0.9228257585i\)
\(L(\frac12)\) \(\approx\) \(3.206055418 + 0.9228257585i\)
\(L(1)\) \(\approx\) \(1.694681484 + 0.5552510442i\)
\(L(1)\) \(\approx\) \(1.694681484 + 0.5552510442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.275 + 0.961i)T \)
3 \( 1 + (0.944 + 0.328i)T \)
5 \( 1 + (0.0875 - 0.996i)T \)
7 \( 1 + (-0.166 - 0.986i)T \)
11 \( 1 + (0.990 - 0.135i)T \)
13 \( 1 + (0.967 - 0.252i)T \)
17 \( 1 + (0.633 - 0.773i)T \)
19 \( 1 + (-0.182 + 0.983i)T \)
23 \( 1 + (0.969 - 0.244i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (0.973 - 0.229i)T \)
37 \( 1 + (-0.213 + 0.976i)T \)
41 \( 1 + (0.283 + 0.959i)T \)
43 \( 1 + (0.947 + 0.321i)T \)
47 \( 1 + (-0.788 - 0.614i)T \)
53 \( 1 + (0.692 + 0.721i)T \)
59 \( 1 + (-0.991 - 0.127i)T \)
61 \( 1 + (-0.460 + 0.887i)T \)
67 \( 1 + (-0.424 + 0.905i)T \)
71 \( 1 + (0.395 - 0.918i)T \)
73 \( 1 + (0.812 + 0.582i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.698 - 0.715i)T \)
89 \( 1 + (-0.516 + 0.856i)T \)
97 \( 1 + (0.166 + 0.986i)T \)
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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

−18.329983848778510532329744635054, −17.83708238058387174313468345833, −17.020068683471194945175930024767, −15.54235675261017733065424876382, −15.31859457998995610247178214743, −14.48099094375188147759687446309, −14.14976722535192485747136568085, −13.380183809333571758518245267626, −12.69036952384969520231577176487, −12.137895744486579552705286564052, −11.26548193796254143425492078997, −10.83264747920804014938918178342, −9.84977055663680281937311480329, −9.164478070034034221757404038708, −8.88293019081686431452635067191, −7.98507777185179818874068612492, −6.96854379195001241143931293565, −6.310584718003242915849612453295, −5.66224988935378971999610647972, −4.50890099860620690098167214583, −3.515724185080342293436568159618, −3.39056783348299563243670760016, −2.3836925791841425813874582187, −1.89364954794046631975023843545, −1.025198017396325480730267852913, 0.8908004016040013681269797289, 1.44443292085033173443332411565, 3.01089162599002323508900390253, 3.63429244313840010634343511060, 4.267742080932450802188066345220, 4.76907933062700712749842061917, 5.75905035955475987060270773289, 6.47287157314130315118115244226, 7.35727853184781822549257986254, 7.97296050037359369900504280243, 8.50549909501330373893683142891, 9.27496168778156152878007319761, 9.66598028177911307063452420183, 10.507952754690201107772240296, 11.63322596627698363372099542341, 12.45782343023208640659121978241, 13.28316532824565793286252234494, 13.53570670561591588047569802109, 14.19868820397466821026515511132, 14.8445919415499558122578176334, 15.542803904451681924738833362871, 16.402813976337385659899919273069, 16.603616168826212280472011851468, 17.1601177714662145246089859264, 18.15103566258565916342533348293

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