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Properties

Label 1-29-29.2-r1-0-0
Degree $1$
Conductor $29$
Sign $0.694 + 0.719i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.694 + 0.719i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ 0.694 + 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.336482735 + 0.9923830322i\)
\(L(\frac12)\) \(\approx\) \(2.336482735 + 0.9923830322i\)
\(L(1)\) \(\approx\) \(1.863111678 + 0.5674444185i\)
\(L(1)\) \(\approx\) \(1.863111678 + 0.5674444185i\)

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

−37.04245994873128389789044407383, −35.53328136868934734040255487507, −34.24374579899933364274784294576, −32.92354006283801009079104039537, −31.686679141284306475076800817427, −30.40092533147660733581688086032, −29.83366549478286309071852784850, −28.65809470427005415901725935009, −26.20135698442904782383539552225, −25.35571381026268010888032397003, −23.88099632625181502635767661730, −22.82998422810975892378699426484, −21.666462301609682914745859493224, −19.75980147859914222535972092754, −19.20336194434541213821538709993, −17.26736173269111935368574092297, −15.13933593332879386040185508097, −14.08824240930443559458423822691, −12.972308637590797923695844339365, −11.59886860374900127592551732455, −9.81363349136584420519654473246, −7.11863534705811442957686118495, −6.387389263564517957755702031095, −3.69799039299330378979350157956, −2.1340826376337406449548185788, 2.955261698120846657748266819765, 4.57621626133495754081804908699, 5.98716994728022731816997731832, 8.38030389475862952098068207369, 9.914179868587829956569145831799, 11.9449356961725847847431895138, 13.29217724059807264511985439907, 14.66857555159666668385681280792, 16.05859785475565978352102140741, 16.77487183832076024476457739420, 19.627794749167200060024001624399, 20.63467801899391642935435791516, 21.83840967026622279248960708327, 22.77741060955489021162623888160, 24.75193013124266946574827913743, 25.29240827090451953382618167494, 26.979572624964354316361891481803, 28.50198626223536218476637900861, 29.74818009913736294213063516409, 31.60940663267064007496181314545, 32.13373860933532408111793833039, 32.93999228210607712199348630910, 34.3737230564112118469584964880, 35.79909569593469150047336538347, 37.52829101282501412682340320577

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