# Properties

 Label 1-104-104.29-r0-0-0 Degree $1$ Conductor $104$ Sign $-0.252 - 0.967i$ Analytic cond. $0.482973$ Root an. cond. $0.482973$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s − 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯
 L(s)  = 1 + (0.5 − 0.866i)3-s − 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.5 + 0.866i)23-s + 25-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$104$$    =    $$2^{3} \cdot 13$$ Sign: $-0.252 - 0.967i$ Analytic conductor: $$0.482973$$ Root analytic conductor: $$0.482973$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{104} (29, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 104,\ (0:\ ),\ -0.252 - 0.967i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5512847262 - 0.7136989931i$$ $$L(\frac12)$$ $$\approx$$ $$0.5512847262 - 0.7136989931i$$ $$L(1)$$ $$\approx$$ $$0.8477874006 - 0.4593680241i$$ $$L(1)$$ $$\approx$$ $$0.8477874006 - 0.4593680241i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$1$$
good3 $$1 + (0.5 - 0.866i)T$$
5 $$1 - T$$
7 $$1 + (-0.5 - 0.866i)T$$
11 $$1 + (0.5 - 0.866i)T$$
17 $$1 + (-0.5 - 0.866i)T$$
19 $$1 + (0.5 + 0.866i)T$$
23 $$1 + (-0.5 + 0.866i)T$$
29 $$1 + (0.5 - 0.866i)T$$
31 $$1 + T$$
37 $$1 + (0.5 - 0.866i)T$$
41 $$1 + (-0.5 + 0.866i)T$$
43 $$1 + (0.5 + 0.866i)T$$
47 $$1 + T$$
53 $$1 - T$$
59 $$1 + (0.5 + 0.866i)T$$
61 $$1 + (0.5 + 0.866i)T$$
67 $$1 + (0.5 - 0.866i)T$$
71 $$1 + (-0.5 - 0.866i)T$$
73 $$1 + T$$
79 $$1 + T$$
83 $$1 - T$$
89 $$1 + (-0.5 + 0.866i)T$$
97 $$1 + (-0.5 - 0.866i)T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−30.565327158432871946955542944338, −28.53443817694127680633050797956, −28.05272845142627925502075510497, −26.97304705150029841955494998337, −26.04454866365307998081401692234, −25.113678981771638114434567172320, −23.8615833924764817968973644715, −22.44563217798018356591253226551, −21.99458759500820733873286531894, −20.46531564833304557727419754431, −19.75268461460064632571256063342, −18.797376279549169811990788189098, −17.22087586641948999326485528498, −15.82815403699149059485070732737, −15.402257400858155073859952808679, −14.3418731883481875157422314124, −12.69949587860829995777604493932, −11.655659553099844315299142300446, −10.35051960445323989587585577542, −9.120690287008776518754663785697, −8.2550868903497222363121493673, −6.70364381300278772246586253413, −4.928118873495006949239879847725, −3.846499361444573578708185295550, −2.53956168451872365177254757974, 0.90263017105314404117282313752, 3.02693120604489576413585062608, 4.07101029658962685950891096778, 6.21450542136112926092012397386, 7.349830454594368796872666702452, 8.19963661788668637866617978427, 9.562476740058880874337850835216, 11.258524197718244437665020164525, 12.148724318318191906642528053062, 13.47105955746550691186765603113, 14.208236028373690216634974205678, 15.66032809807942426928061042229, 16.690889637544991192160021246761, 18.05767607010868325139665588364, 19.260978558693066293949714776853, 19.73439057130018001173306386502, 20.77973713847720101505492447631, 22.552500994426656066777050993697, 23.34782178057327352095367175836, 24.27744859916826801611122737905, 25.198926543745115301679106040975, 26.59974403815828800238010900459, 27.0218468261073436319733812201, 28.64689773544053930107950935234, 29.705407270134012978408597291778