Properties

Label 1-4729-4729.2986-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.741 + 0.671i$
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  + (0.999 + 0.0318i)2-s + (−0.830 − 0.556i)3-s + (0.997 + 0.0637i)4-s + (−0.996 − 0.0796i)5-s + (−0.812 − 0.582i)6-s + (0.290 + 0.956i)7-s + (0.995 + 0.0955i)8-s + (0.380 + 0.924i)9-s + (−0.993 − 0.111i)10-s + (−0.742 − 0.669i)11-s + (−0.793 − 0.608i)12-s + (−0.366 − 0.930i)13-s + (0.260 + 0.965i)14-s + (0.783 + 0.620i)15-s + (0.991 + 0.127i)16-s + (0.614 + 0.788i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0318i)2-s + (−0.830 − 0.556i)3-s + (0.997 + 0.0637i)4-s + (−0.996 − 0.0796i)5-s + (−0.812 − 0.582i)6-s + (0.290 + 0.956i)7-s + (0.995 + 0.0955i)8-s + (0.380 + 0.924i)9-s + (−0.993 − 0.111i)10-s + (−0.742 − 0.669i)11-s + (−0.793 − 0.608i)12-s + (−0.366 − 0.930i)13-s + (0.260 + 0.965i)14-s + (0.783 + 0.620i)15-s + (0.991 + 0.127i)16-s + (0.614 + 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.741 + 0.671i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2986, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.741 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2559045152 + 0.6639918564i\)
\(L(\frac12)\) \(\approx\) \(0.2559045152 + 0.6639918564i\)
\(L(1)\) \(\approx\) \(1.110759273 + 0.03203538044i\)
\(L(1)\) \(\approx\) \(1.110759273 + 0.03203538044i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.999 + 0.0318i)T \)
3 \( 1 + (-0.830 - 0.556i)T \)
5 \( 1 + (-0.996 - 0.0796i)T \)
7 \( 1 + (0.290 + 0.956i)T \)
11 \( 1 + (-0.742 - 0.669i)T \)
13 \( 1 + (-0.366 - 0.930i)T \)
17 \( 1 + (0.614 + 0.788i)T \)
19 \( 1 + (-0.0239 + 0.999i)T \)
23 \( 1 + (-0.872 + 0.488i)T \)
29 \( 1 + (-0.999 + 0.0318i)T \)
31 \( 1 + (0.709 - 0.704i)T \)
37 \( 1 + (0.495 + 0.868i)T \)
41 \( 1 + (-0.166 + 0.986i)T \)
43 \( 1 + (0.663 - 0.748i)T \)
47 \( 1 + (-0.967 + 0.252i)T \)
53 \( 1 + (-0.901 - 0.431i)T \)
59 \( 1 + (-0.563 - 0.826i)T \)
61 \( 1 + (0.275 + 0.961i)T \)
67 \( 1 + (-0.921 - 0.388i)T \)
71 \( 1 + (-0.589 - 0.808i)T \)
73 \( 1 + (0.410 - 0.912i)T \)
79 \( 1 - T \)
83 \( 1 + (0.549 + 0.835i)T \)
89 \( 1 + (0.978 + 0.205i)T \)
97 \( 1 + (0.290 + 0.956i)T \)
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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

−17.71253351048583935237619474838, −17.030384748593109721113306536646, −16.16219843855650633660380744099, −16.07722037550101548719638622597, −15.259016241743855657864745771776, −14.53292493431110982552638931030, −14.069332188483575314250719755710, −13.02204103576076873412035062453, −12.46812169950302751846374901873, −11.69256581977533001259816177287, −11.37397188980699666535152794244, −10.62132220479724853714170505064, −10.10712351244017845178563753055, −9.21260828509679314197035122694, −7.91966975636180709985281672079, −7.2560895732047006411688015130, −6.94692946987578324132801881281, −5.99510556873087344805263822966, −4.96068297757718999534647012082, −4.64008856405024055167320413777, −4.10015920547706494801788520629, −3.33930062273620824800163500083, −2.41470904147133513471040797597, −1.25421861032178334150952456574, −0.15330590061636819989539125628, 1.20578110601744375242249470502, 2.051106186653640082630272789544, 2.982786455616031377112954378952, 3.63189586725115378364381339617, 4.65458322735693488731725357146, 5.22771339096086410802927906912, 5.973108250241685314651245617722, 6.21510434593501745221172596096, 7.638888366984624716027878367145, 7.81953432434736219903496964954, 8.32912383837738958963872896782, 9.906577418720234953171446571537, 10.59756230509978061237064262119, 11.306086177483327306358933917616, 11.79959690099454301059873294048, 12.37808393678052353358163605133, 12.84310709297615190240465511929, 13.49584608892013218179529271736, 14.511832292320485235289257728045, 15.1241390431631344465844884307, 15.63396358301169353584804013812, 16.35546898799075490637248511646, 16.80377098157945768615974949961, 17.7595102678029378114483280894, 18.55574259590924789515125426792

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