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Properties

Label 1-29-29.18-r1-0-0
Degree $1$
Conductor $29$
Sign $0.748 - 0.663i$
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  + (−0.781 + 0.623i)2-s + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.433 + 0.900i)8-s + (0.900 − 0.433i)9-s + (0.974 + 0.222i)10-s + (0.433 − 0.900i)11-s − i·12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)14-s + (−0.781 − 0.623i)15-s + (−0.900 − 0.433i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.433 + 0.900i)8-s + (0.900 − 0.433i)9-s + (0.974 + 0.222i)10-s + (0.433 − 0.900i)11-s − i·12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)14-s + (−0.781 − 0.623i)15-s + (−0.900 − 0.433i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.748 - 0.663i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ 0.748 - 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087081239 - 0.4123738032i\)
\(L(\frac12)\) \(\approx\) \(1.087081239 - 0.4123738032i\)
\(L(1)\) \(\approx\) \(0.9517205724 - 0.1276036800i\)
\(L(1)\) \(\approx\) \(0.9517205724 - 0.1276036800i\)

Euler product

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

−37.46986994404126658566257533250, −35.94063821357781366743636405717, −35.08979317042307321814473340728, −33.61056647325815667431121405335, −31.64937470322825613604807963353, −30.86154201128395854661805431599, −29.826163799698810649399415439403, −27.97443704409467097341639975847, −27.21136163695190757224101260428, −25.782990453369060922239324232262, −25.18560929464432528653516989071, −22.72615278105629039279741289762, −21.40393052872404783201839018658, −20.122142224187477977562533148425, −19.04373480829460525105877112993, −18.13045620199102475244654139509, −15.940840478562203687453697352655, −14.89116771884443934774591593765, −12.88744644711863407398966621349, −11.348964611897701878111886519811, −9.785296740604278280112876041834, −8.55162982325935521455889507174, −7.15117898661048565153829143194, −3.746600394393176169284577965833, −2.3662445182461734522183289831, 1.108006771234075177172717499301, 4.032093181053488653242247587450, 6.63009378732893884977654244674, 8.15878852533203210536033250729, 8.99537501336799823116965325086, 10.82428153570607947611710280105, 13.119715830924541798111648536269, 14.48174747157975039406996363439, 15.954688811106935113934370831995, 17.0044815211619784788108709693, 18.944522888903973775404159782323, 19.70135093836385893835531792831, 20.86140831918805662796470377969, 23.525010012698904818179661474238, 24.22176112881330084964965045254, 25.60382161244306762187339201257, 26.63842281989049923602423545720, 27.65574336461099084076831662253, 29.208314400232032490220014496006, 30.696356685737118594815641196403, 32.341501795118824234035018108792, 32.79690552463707388910807018314, 34.78843057746584802474589897246, 35.844377000987124402476042857646, 36.50020029788882475712826055811

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