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Properties

Label 1-4729-4729.91-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.994 + 0.101i$
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  + (0.563 − 0.826i)2-s + (0.753 − 0.657i)3-s + (−0.366 − 0.930i)4-s + (0.982 + 0.184i)5-s + (−0.119 − 0.992i)6-s + (0.986 − 0.164i)7-s + (−0.975 − 0.221i)8-s + (0.135 − 0.990i)9-s + (0.706 − 0.708i)10-s + (−0.373 − 0.927i)11-s + (−0.887 − 0.460i)12-s + (−0.767 − 0.641i)13-s + (0.419 − 0.907i)14-s + (0.862 − 0.506i)15-s + (−0.732 + 0.681i)16-s + (0.0265 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.563 − 0.826i)2-s + (0.753 − 0.657i)3-s + (−0.366 − 0.930i)4-s + (0.982 + 0.184i)5-s + (−0.119 − 0.992i)6-s + (0.986 − 0.164i)7-s + (−0.975 − 0.221i)8-s + (0.135 − 0.990i)9-s + (0.706 − 0.708i)10-s + (−0.373 − 0.927i)11-s + (−0.887 − 0.460i)12-s + (−0.767 − 0.641i)13-s + (0.419 − 0.907i)14-s + (0.862 − 0.506i)15-s + (−0.732 + 0.681i)16-s + (0.0265 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.994 + 0.101i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.994 + 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2114557362 - 4.147443679i\)
\(L(\frac12)\) \(\approx\) \(-0.2114557362 - 4.147443679i\)
\(L(1)\) \(\approx\) \(1.270712186 - 1.770037782i\)
\(L(1)\) \(\approx\) \(1.270712186 - 1.770037782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.563 - 0.826i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
5 \( 1 + (0.982 + 0.184i)T \)
7 \( 1 + (0.986 - 0.164i)T \)
11 \( 1 + (-0.373 - 0.927i)T \)
13 \( 1 + (-0.767 - 0.641i)T \)
17 \( 1 + (0.0265 - 0.999i)T \)
19 \( 1 + (0.892 - 0.450i)T \)
23 \( 1 + (-0.143 - 0.989i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (-0.267 + 0.963i)T \)
37 \( 1 + (0.935 + 0.353i)T \)
41 \( 1 + (0.924 + 0.380i)T \)
43 \( 1 + (-0.920 + 0.390i)T \)
47 \( 1 + (0.560 + 0.827i)T \)
53 \( 1 + (-0.863 + 0.504i)T \)
59 \( 1 + (-0.643 + 0.765i)T \)
61 \( 1 + (-0.922 - 0.385i)T \)
67 \( 1 + (-0.597 - 0.801i)T \)
71 \( 1 + (0.994 + 0.100i)T \)
73 \( 1 + (-0.0610 - 0.998i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.977 + 0.211i)T \)
89 \( 1 + (0.534 + 0.845i)T \)
97 \( 1 + (-0.986 + 0.164i)T \)
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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

−18.16717537074726138563509937557, −17.727815454568991667080379587606, −17.00213287575186579791672460283, −16.52285013000407141912790539297, −15.61990990312509292285211265181, −15.01538314231327231350154431834, −14.52919764079360400313591031903, −14.02189988583412150217965343468, −13.425383432955833773686421934624, −12.64453828216450897115077008633, −11.98661358473217564089040315136, −11.02407410292230719257579637904, −10.08663043694217706152216671365, −9.50922682305170227935346834997, −8.96610880580067313755617894851, −8.05820440646768305514193495603, −7.64428677145675446616237451866, −6.847575271544070241826953733548, −5.72706204176606329991454787967, −5.2772114863522829815378874385, −4.59244394750471184998639773830, −4.05270820272555283438285930332, −3.00936154799910183223973840987, −2.160705307616624129661045595956, −1.65429821401179405880778394192, 0.79240940219997387401533291508, 1.25720421018555459029522355227, 2.38490287367168655042141809792, 2.70628008086331747349400743405, 3.31548837863403719102009741444, 4.65017278237558965756731530048, 5.0417797580326151669262690213, 5.96343048031606475871811141046, 6.58902764845600379278843601410, 7.60711261946957328487162732118, 8.203485327716072339288400263847, 9.19507122992751706004653005174, 9.54420515013142361127051290979, 10.52300833777359531540344954184, 10.98285804405188225685104121067, 11.92982407949922967761131487872, 12.42934527147712732237024039901, 13.31779476949421582458550654601, 13.794222892121721108136786427757, 14.15648196270191370515220345160, 14.76104156844953151944683359788, 15.467405377350075404107816338574, 16.506836232433190196638332917891, 17.59003835649277604534493105213, 18.10045199745629365384524164743

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