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Properties

Label 1-4729-4729.149-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.867 - 0.496i$
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  + (−0.336 + 0.941i)2-s + (−0.663 − 0.748i)3-s + (−0.773 − 0.633i)4-s + (0.827 + 0.560i)5-s + (0.927 − 0.373i)6-s + (−0.994 − 0.100i)7-s + (0.856 − 0.516i)8-s + (−0.119 + 0.992i)9-s + (−0.806 + 0.591i)10-s + (−0.999 − 0.0318i)11-s + (0.0398 + 0.999i)12-s + (0.0292 − 0.999i)13-s + (0.429 − 0.903i)14-s + (−0.129 − 0.991i)15-s + (0.198 + 0.980i)16-s + (−0.996 − 0.0849i)17-s + ⋯
L(s)  = 1  + (−0.336 + 0.941i)2-s + (−0.663 − 0.748i)3-s + (−0.773 − 0.633i)4-s + (0.827 + 0.560i)5-s + (0.927 − 0.373i)6-s + (−0.994 − 0.100i)7-s + (0.856 − 0.516i)8-s + (−0.119 + 0.992i)9-s + (−0.806 + 0.591i)10-s + (−0.999 − 0.0318i)11-s + (0.0398 + 0.999i)12-s + (0.0292 − 0.999i)13-s + (0.429 − 0.903i)14-s + (−0.129 − 0.991i)15-s + (0.198 + 0.980i)16-s + (−0.996 − 0.0849i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.867 - 0.496i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.867 - 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4118915730 - 0.1095817514i\)
\(L(\frac12)\) \(\approx\) \(0.4118915730 - 0.1095817514i\)
\(L(1)\) \(\approx\) \(0.5319077486 + 0.1243303930i\)
\(L(1)\) \(\approx\) \(0.5319077486 + 0.1243303930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.336 + 0.941i)T \)
3 \( 1 + (-0.663 - 0.748i)T \)
5 \( 1 + (0.827 + 0.560i)T \)
7 \( 1 + (-0.994 - 0.100i)T \)
11 \( 1 + (-0.999 - 0.0318i)T \)
13 \( 1 + (0.0292 - 0.999i)T \)
17 \( 1 + (-0.996 - 0.0849i)T \)
19 \( 1 + (-0.925 - 0.378i)T \)
23 \( 1 + (-0.698 - 0.715i)T \)
29 \( 1 + (0.336 + 0.941i)T \)
31 \( 1 + (0.971 + 0.236i)T \)
37 \( 1 + (-0.746 + 0.665i)T \)
41 \( 1 + (0.812 + 0.582i)T \)
43 \( 1 + (0.824 + 0.565i)T \)
47 \( 1 + (-0.796 + 0.604i)T \)
53 \( 1 + (-0.906 + 0.422i)T \)
59 \( 1 + (0.962 - 0.272i)T \)
61 \( 1 + (-0.999 - 0.0106i)T \)
67 \( 1 + (-0.986 + 0.164i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (-0.679 + 0.733i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.777 + 0.629i)T \)
89 \( 1 + (0.385 - 0.922i)T \)
97 \( 1 + (-0.994 - 0.100i)T \)
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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

−17.9788975494643327427141554034, −17.63886379768424974506195490711, −17.020479234778457059573470302202, −16.20203126477361329742994502192, −15.95865779190398031146693026076, −14.95457792931846757792592975272, −13.75367667802679658183938137060, −13.489084322274910107012543207267, −12.58668936630349503648910133003, −12.20158907439721455173432749968, −11.36367103559162459978020049316, −10.56268542781330086707564695526, −10.121138213834113288798285458893, −9.529177257045311977580880760706, −8.97285528828082653552087854536, −8.35073500276488894382085440827, −7.13629270733042228819041883647, −6.1608389570997452425306161341, −5.7294217308406572125666123602, −4.648748106795937788676023345017, −4.293614582757749530859070670318, −3.40704482418197558841690828624, −2.39649057353149099658566979303, −1.88466695049836027311089378408, −0.55202106787465346228051826448, 0.25095298931046080580240315734, 1.32154295972381904363339520926, 2.43402206908428997780733954535, 2.990583586433961044448425339534, 4.4698517088539191699102370187, 5.12251211172845897039871361698, 5.976578480088459079286754467128, 6.36611935701155142277348740674, 6.83467781345032148806887842156, 7.66951127289920414954793626948, 8.32582065938595261584808291880, 9.15671131165994172812008139129, 10.08553059209938837497131428292, 10.54386566068193625080699866624, 10.9419733682172705380796227100, 12.3695902010032511913930578392, 13.0518360331641741342272896508, 13.25993345295909052381658152573, 14.017678722924216199209941665107, 14.80120582983459241383250216574, 15.75246726172580318766514764295, 15.99919486879174902038953424940, 16.94190075745434983434308331216, 17.51191169403182540709498524096, 18.03997393176553620526985086564

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