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Properties

Label 1-4729-4729.251-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.345 - 0.938i$
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  + (0.981 − 0.190i)2-s + (−0.921 + 0.388i)3-s + (0.927 − 0.373i)4-s + (0.887 − 0.460i)5-s + (−0.830 + 0.556i)6-s + (0.198 − 0.980i)7-s + (0.839 − 0.543i)8-s + (0.698 − 0.715i)9-s + (0.783 − 0.620i)10-s + (−0.305 + 0.952i)11-s + (−0.709 + 0.704i)12-s + (0.627 − 0.778i)13-s + (0.00797 − 0.999i)14-s + (−0.639 + 0.768i)15-s + (0.721 − 0.692i)16-s + (0.675 + 0.737i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.190i)2-s + (−0.921 + 0.388i)3-s + (0.927 − 0.373i)4-s + (0.887 − 0.460i)5-s + (−0.830 + 0.556i)6-s + (0.198 − 0.980i)7-s + (0.839 − 0.543i)8-s + (0.698 − 0.715i)9-s + (0.783 − 0.620i)10-s + (−0.305 + 0.952i)11-s + (−0.709 + 0.704i)12-s + (0.627 − 0.778i)13-s + (0.00797 − 0.999i)14-s + (−0.639 + 0.768i)15-s + (0.721 − 0.692i)16-s + (0.675 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.819663032 - 1.965834912i\)
\(L(\frac12)\) \(\approx\) \(2.819663032 - 1.965834912i\)
\(L(1)\) \(\approx\) \(1.803249583 - 0.5109118419i\)
\(L(1)\) \(\approx\) \(1.803249583 - 0.5109118419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.981 - 0.190i)T \)
3 \( 1 + (-0.921 + 0.388i)T \)
5 \( 1 + (0.887 - 0.460i)T \)
7 \( 1 + (0.198 - 0.980i)T \)
11 \( 1 + (-0.305 + 0.952i)T \)
13 \( 1 + (0.627 - 0.778i)T \)
17 \( 1 + (0.675 + 0.737i)T \)
19 \( 1 + (-0.989 + 0.143i)T \)
23 \( 1 + (-0.996 + 0.0796i)T \)
29 \( 1 + (0.981 + 0.190i)T \)
31 \( 1 + (-0.0239 - 0.999i)T \)
37 \( 1 + (0.999 - 0.0318i)T \)
41 \( 1 + (-0.536 + 0.843i)T \)
43 \( 1 + (0.351 - 0.936i)T \)
47 \( 1 + (0.0398 + 0.999i)T \)
53 \( 1 + (-0.894 - 0.446i)T \)
59 \( 1 + (0.901 + 0.431i)T \)
61 \( 1 + (0.103 - 0.994i)T \)
67 \( 1 + (-0.732 - 0.681i)T \)
71 \( 1 + (0.803 + 0.595i)T \)
73 \( 1 + (0.821 + 0.569i)T \)
79 \( 1 + T \)
83 \( 1 + (0.939 + 0.343i)T \)
89 \( 1 + (0.321 - 0.947i)T \)
97 \( 1 + (0.198 - 0.980i)T \)
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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

−18.10132090928675521112017540216, −17.74725928465936903131813797935, −16.70383273991214815843968778682, −16.29646142999292426595692163107, −15.72043490270238787399908044657, −14.8124078028046261632620227913, −14.07977829475440410174407809033, −13.63550514660079486599995717581, −12.986513625591101791942203511624, −12.11638619014324872910049309429, −11.78809422235316257115955709733, −10.91899115066708090017517322662, −10.56393186268967938111806650840, −9.547103613457093126022677148241, −8.52316990640399260730911475924, −7.86464848583976834004337589894, −6.75939250366045892679597260317, −6.37619867660081953499927092970, −5.81618331509863348570827852669, −5.26368092051110739646949133768, −4.55750038670417598384919950933, −3.49205098238242162711435594210, −2.56207443718208384055500652732, −2.0488331486911448148409396930, −1.13990223879356303131002840468, 0.75668472609342506014315677813, 1.506269194568042766899791648412, 2.26947298820325346371371336971, 3.495699375340502449403773569575, 4.19475496963087167065936977633, 4.719390997097334055308854891980, 5.41619207219121046748173350106, 6.19886610761186191764407598486, 6.472554136823744403312525622503, 7.57879720819766098934387899805, 8.24448044676669777555667747914, 9.76109684299192387890462650654, 9.999296261839480067535514892154, 10.65081575866214944789565836848, 11.15504015438925658278542620214, 12.21141589605674521970653358288, 12.68480590582352148087896121928, 13.14515545394349673669049103655, 13.91816936389275829721875699747, 14.67651427055755468493198317937, 15.281959119768192875170800030321, 16.06498236417137830351224565784, 16.67293729486622769124639323071, 17.2585078656496754729333187622, 17.77095053486170065719373640511

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