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Properties

Label 1-4729-4729.71-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.655 + 0.754i$
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 + (−0.906 + 0.422i)5-s + (−0.0557 + 0.998i)6-s + (−0.770 + 0.637i)7-s + (−0.742 − 0.669i)8-s + (0.783 + 0.620i)9-s + (−0.655 − 0.755i)10-s + (0.990 − 0.135i)11-s + (−0.975 + 0.221i)12-s + (−0.702 − 0.711i)13-s + (−0.824 − 0.565i)14-s + (−0.994 + 0.100i)15-s + (0.439 − 0.898i)16-s + (−0.986 − 0.161i)17-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (0.944 + 0.328i)3-s + (−0.848 + 0.529i)4-s + (−0.906 + 0.422i)5-s + (−0.0557 + 0.998i)6-s + (−0.770 + 0.637i)7-s + (−0.742 − 0.669i)8-s + (0.783 + 0.620i)9-s + (−0.655 − 0.755i)10-s + (0.990 − 0.135i)11-s + (−0.975 + 0.221i)12-s + (−0.702 − 0.711i)13-s + (−0.824 − 0.565i)14-s + (−0.994 + 0.100i)15-s + (0.439 − 0.898i)16-s + (−0.986 − 0.161i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.655 + 0.754i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.655 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8144440265 + 1.786301441i\)
\(L(\frac12)\) \(\approx\) \(0.8144440265 + 1.786301441i\)
\(L(1)\) \(\approx\) \(0.9266977547 + 0.8449035226i\)
\(L(1)\) \(\approx\) \(0.9266977547 + 0.8449035226i\)

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 \( 1 + (-0.906 + 0.422i)T \)
7 \( 1 + (-0.770 + 0.637i)T \)
11 \( 1 + (0.990 - 0.135i)T \)
13 \( 1 + (-0.702 - 0.711i)T \)
17 \( 1 + (-0.986 - 0.161i)T \)
19 \( 1 + (0.942 - 0.333i)T \)
23 \( 1 + (0.969 - 0.244i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (0.973 - 0.229i)T \)
37 \( 1 + (0.952 - 0.303i)T \)
41 \( 1 + (0.283 + 0.959i)T \)
43 \( 1 + (-0.751 + 0.659i)T \)
47 \( 1 + (-0.137 + 0.990i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.606 - 0.795i)T \)
61 \( 1 + (-0.538 - 0.842i)T \)
67 \( 1 + (-0.571 - 0.820i)T \)
71 \( 1 + (0.597 + 0.801i)T \)
73 \( 1 + (0.0981 - 0.995i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.270 + 0.962i)T \)
89 \( 1 + (0.999 + 0.0186i)T \)
97 \( 1 + (0.770 - 0.637i)T \)
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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

−18.22403217585019353022178048334, −17.256114261041098575410088562, −16.659946048212520626886791725841, −15.7261170085669728137857117478, −15.00534759589461013238746462474, −14.484730187521123580286650964710, −13.65702633231153090426085801338, −13.233523865922790491057440145274, −12.53645735187147340908994885138, −11.880312559501095118227115639329, −11.42344220326235294690675859524, −10.3790389310413654706636063013, −9.64479390272024952537522420809, −9.07258109940427386492808357671, −8.67443335113806782122365101732, −7.538943176146341414544629084538, −7.05403801713114294342893090639, −6.25251800185645513860303398941, −4.91182346594666617123911328471, −4.31679086982178812892365274100, −3.660422122612741949571278220371, −3.233857660123523754864215374986, −2.239137398323467593614612993828, −1.38755410001236602354931349407, −0.63616722011611268657908698239, 0.74300903370157771241255085451, 2.41592107549880066797172098934, 3.1562166152628097493704719469, 3.4939606475224867561933611151, 4.539036759604805153831957648841, 4.912533268615699567232189498351, 6.19668144225416917147768825067, 6.68066546506232976839263943496, 7.5138881145625647424571007517, 7.959482329468421073254234923248, 8.7947756342395157352661500892, 9.42141978249133119522670368321, 9.77328419494698529664281840409, 11.047431608036846538335160215788, 11.73833524751264848687849089861, 12.69595506035060568691550855086, 13.04460875291957450242060506568, 13.95293531816893874625599848795, 14.590861940348169927450558272460, 15.26365606451689887982855068577, 15.40359286516657287884128616601, 16.20768413982537736550576740381, 16.75731674423764917621312907892, 17.71529402894905326431147517227, 18.48120661944581856984474931408

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