# Properties

 Label 1-29-29.17-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 $0$

# Related objects

## 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: $\chi_{29} (17, \cdot )$ 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

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