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Properties

Label 1-4729-4729.643-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.726 - 0.687i$
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  + (−0.830 − 0.556i)2-s + (−0.996 + 0.0796i)3-s + (0.380 + 0.924i)4-s + (0.995 − 0.0955i)5-s + (0.872 + 0.488i)6-s + (0.0398 − 0.999i)7-s + (0.198 − 0.980i)8-s + (0.987 − 0.158i)9-s + (−0.880 − 0.474i)10-s + (0.536 + 0.843i)11-s + (−0.453 − 0.891i)12-s + (0.135 − 0.990i)13-s + (−0.589 + 0.808i)14-s + (−0.984 + 0.174i)15-s + (−0.709 + 0.704i)16-s + (0.894 − 0.446i)17-s + ⋯
L(s)  = 1  + (−0.830 − 0.556i)2-s + (−0.996 + 0.0796i)3-s + (0.380 + 0.924i)4-s + (0.995 − 0.0955i)5-s + (0.872 + 0.488i)6-s + (0.0398 − 0.999i)7-s + (0.198 − 0.980i)8-s + (0.987 − 0.158i)9-s + (−0.880 − 0.474i)10-s + (0.536 + 0.843i)11-s + (−0.453 − 0.891i)12-s + (0.135 − 0.990i)13-s + (−0.589 + 0.808i)14-s + (−0.984 + 0.174i)15-s + (−0.709 + 0.704i)16-s + (0.894 − 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3743526332 - 0.9397649593i\)
\(L(\frac12)\) \(\approx\) \(0.3743526332 - 0.9397649593i\)
\(L(1)\) \(\approx\) \(0.6367888115 - 0.3231474479i\)
\(L(1)\) \(\approx\) \(0.6367888115 - 0.3231474479i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.830 - 0.556i)T \)
3 \( 1 + (-0.996 + 0.0796i)T \)
5 \( 1 + (0.995 - 0.0955i)T \)
7 \( 1 + (0.0398 - 0.999i)T \)
11 \( 1 + (0.536 + 0.843i)T \)
13 \( 1 + (0.135 - 0.990i)T \)
17 \( 1 + (0.894 - 0.446i)T \)
19 \( 1 + (-0.336 - 0.941i)T \)
23 \( 1 + (0.999 - 0.0159i)T \)
29 \( 1 + (0.830 - 0.556i)T \)
31 \( 1 + (-0.949 + 0.313i)T \)
37 \( 1 + (-0.812 - 0.582i)T \)
41 \( 1 + (-0.675 + 0.737i)T \)
43 \( 1 + (-0.0717 + 0.997i)T \)
47 \( 1 + (-0.00797 - 0.999i)T \)
53 \( 1 + (0.395 - 0.918i)T \)
59 \( 1 + (-0.753 - 0.657i)T \)
61 \( 1 + (0.944 + 0.328i)T \)
67 \( 1 + (0.887 - 0.460i)T \)
71 \( 1 + (0.991 + 0.127i)T \)
73 \( 1 + (-0.732 - 0.681i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (0.639 - 0.768i)T \)
97 \( 1 + (0.0398 - 0.999i)T \)
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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

−18.55818094222021531639509238268, −17.572592861546363924391019165171, −17.039214052757050490579907495721, −16.65671408663160260484637312749, −16.02104017775691968000653783223, −15.23160437087346290000576170501, −14.39150616816307034492506514853, −14.01037242339613507581201873421, −12.94629653221501433204841631012, −12.152093915908120606603127588015, −11.585643883136483147039391899597, −10.766224446642931308001172902387, −10.29981768025471060090473326961, −9.46727250848122706477805580496, −8.91763397249028153357418264080, −8.31451505887276502934303222823, −7.15785189677913519881647671490, −6.58218172509127033045893016538, −5.97175209246481107816218523207, −5.53787441599080563536500302995, −4.92153855367751777792190221249, −3.65853351361503484684697972226, −2.45456508190946850516005200420, −1.55795983120786192429660642665, −1.18010618597298641254745517667, 0.47287190010130392275041885276, 1.165174881704426849951093315905, 1.8255535037517948687191783807, 2.91799690948729368900388811746, 3.73455370194157432167991926574, 4.727002170339138348113747153743, 5.23451894557884012225503168416, 6.35860510808177036045845267002, 6.939661344030263082715736370391, 7.40437127642038873091774708111, 8.438844541223006985208565511464, 9.342633294211687499717424406440, 10.006976236074696718602214127995, 10.25090570841155374604055026457, 11.039074285295292162539056097222, 11.5802244827010236641758700368, 12.61113205645184917483844405033, 12.86727201158520371866144045533, 13.597243242590990421061007020755, 14.584397075562706508913136650101, 15.45903578780832347491675125049, 16.26868704250822402037721896768, 16.98855087903774994535877075713, 17.20319312140972873761550559487, 17.89556896510534492595576923472

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