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Properties

Label 1-29-29.12-r1-0-0
Degree $1$
Conductor $29$
Sign $-0.981 - 0.189i$
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  + i·2-s + i·3-s − 4-s − 5-s − 6-s + 7-s − i·8-s − 9-s − i·10-s + i·11-s − i·12-s − 13-s + i·14-s − i·15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 5-s − 6-s + 7-s − i·8-s − 9-s − i·10-s + i·11-s − i·12-s − 13-s + i·14-s − i·15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ -0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08474782205 + 0.8882062763i\)
\(L(\frac12)\) \(\approx\) \(-0.08474782205 + 0.8882062763i\)
\(L(1)\) \(\approx\) \(0.4625314657 + 0.6831742205i\)
\(L(1)\) \(\approx\) \(0.4625314657 + 0.6831742205i\)

Euler product

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

−36.58119942230428228497938358477, −35.288376310747336528673827379967, −34.30237247785495133662555359237, −31.93508970853058496743106932218, −31.10033503937721751946229597796, −30.12616569043735878056927390136, −29.072204485983874937780182997194, −27.61921386608848623556906906964, −26.68504872206176706946738866373, −24.45661856878935023919828479746, −23.59699895109500485861873311974, −22.21796675518490570516333527597, −20.5433045039491820431371102294, −19.41065511827340023321975207467, −18.49087283075723871829389208359, −17.12410967854378792813736289340, −14.69959539671979915787085416335, −13.36551898131251656190980181990, −11.83682365408827654656604081239, −11.2046802679650549553576079672, −8.752860374100181475943392429568, −7.522370018623240236906305643369, −4.92154672458480704670784048627, −2.79425321821631066872263286931, −0.68477668684036983552862578190, 4.05675122682315940902913033802, 5.13369232396478138803645339201, 7.43927611978010320370504802554, 8.69691382994615739673561884125, 10.39481769083959126746112127147, 12.219817511718669520572046504276, 14.64818280991473529335445209758, 15.074498237549355943996857681023, 16.536379178954270945798445322585, 17.667534009598030427717159003844, 19.54717863875444077217495449992, 21.13779486925325576335559493554, 22.599167521713952286715636391845, 23.58984753725603607550547188137, 25.00640919340602308662924960263, 26.5464547763482008640984614059, 27.305931348849879430394125483881, 28.1911238457046417003486176760, 30.8358564303758558294315479574, 31.57018795331613411475634102548, 32.90741085263964068537639021561, 33.93502480075039901077519180499, 34.793337516997726926932590110811, 36.29709377351021840681713540575, 37.5288509908176280059873365156

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