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Properties

Label 1-29-29.14-r1-0-0
Degree $1$
Conductor $29$
Sign $-0.965 - 0.259i$
Analytic cond. $3.11648$
Root an. cond. $3.11648$
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  + (−0.974 + 0.222i)2-s + (−0.433 + 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s + i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s − i·17-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (−0.433 + 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s + i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $-0.965 - 0.259i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ -0.965 - 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03893022907 + 0.2945361561i\)
\(L(\frac12)\) \(\approx\) \(-0.03893022907 + 0.2945361561i\)
\(L(1)\) \(\approx\) \(0.3747250616 + 0.2511309087i\)
\(L(1)\) \(\approx\) \(0.3747250616 + 0.2511309087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
3 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.433 + 0.900i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.974 + 0.222i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 + (0.781 - 0.623i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.974 + 0.222i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−36.32884897925973751989944978013, −35.2287473100929020355826505757, −34.45580871507419549027526719744, −32.73570361008079025482551567695, −31.0516154965372715801311961486, −29.61490271416393009660347178927, −28.62418795508065674265655458023, −28.05554372004965076718262632396, −26.01858014764433045097600224549, −25.01209183772276175636572737841, −23.95026749090851111165224916931, −22.13517852382507452923648244378, −20.35801547798586845634103174551, −19.34121122079636210907478682492, −17.99049740616847144593320782810, −16.99562651617503798887169364400, −15.6912475455025026781435499883, −12.881093689766006580809061459344, −12.335457616553437763722478223536, −10.384437039107422299995041945529, −8.817102143627068953062667097247, −7.37199325865796530868825297488, −5.69474229033383494236138712733, −2.30697289594641798048655483506, −0.2862934905105097622415294941, 3.11052327318947713514380790737, 5.81118686132454743263652222646, 7.236135164049806406728716369542, 9.45973420770702444627571995304, 10.32166071270391729715886245053, 11.548331018847100495150821730577, 14.22083851511268109240505224622, 15.747003021916387875952157815752, 16.62969732897592530304161592763, 18.08164008228776669357575914508, 19.32358314422320525374585276070, 20.898215721725029759565808195954, 22.278173584955518879505979603704, 23.605109310255890352744603238731, 25.59789200548448145166254240641, 26.52134535464006100913263987891, 27.226945801886639338215726681246, 29.05268673290717849849720695740, 29.38735919294798782692708859755, 31.71972518357945938458926289390, 33.262254115807369595485829812974, 33.943576809860669899419047960466, 35.008586385972894814041117190877, 36.46960502124592209044403414400, 37.80993381257266007001975821627

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