# Properties

 Label 1-916-916.703-r1-0-0 Degree $1$ Conductor $916$ Sign $-0.532 + 0.846i$ Analytic cond. $98.4378$ Root an. cond. $98.4378$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.677 + 0.735i)3-s + (0.789 + 0.614i)5-s + (0.879 − 0.475i)7-s + (−0.0825 + 0.996i)9-s + (0.401 + 0.915i)11-s + (0.789 + 0.614i)13-s + (0.0825 + 0.996i)15-s + (0.789 + 0.614i)17-s + (−0.789 + 0.614i)19-s + (0.945 + 0.324i)21-s + (−0.546 − 0.837i)23-s + (0.245 + 0.969i)25-s + (−0.789 + 0.614i)27-s + (−0.879 + 0.475i)29-s + (0.401 + 0.915i)31-s + ⋯
 L(s)  = 1 + (0.677 + 0.735i)3-s + (0.789 + 0.614i)5-s + (0.879 − 0.475i)7-s + (−0.0825 + 0.996i)9-s + (0.401 + 0.915i)11-s + (0.789 + 0.614i)13-s + (0.0825 + 0.996i)15-s + (0.789 + 0.614i)17-s + (−0.789 + 0.614i)19-s + (0.945 + 0.324i)21-s + (−0.546 − 0.837i)23-s + (0.245 + 0.969i)25-s + (−0.789 + 0.614i)27-s + (−0.879 + 0.475i)29-s + (0.401 + 0.915i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$916$$    =    $$2^{2} \cdot 229$$ Sign: $-0.532 + 0.846i$ Analytic conductor: $$98.4378$$ Root analytic conductor: $$98.4378$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{916} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 916,\ (1:\ ),\ -0.532 + 0.846i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.883727301 + 3.411790829i$$ $$L(\frac12)$$ $$\approx$$ $$1.883727301 + 3.411790829i$$ $$L(1)$$ $$\approx$$ $$1.555781461 + 0.8933196916i$$ $$L(1)$$ $$\approx$$ $$1.555781461 + 0.8933196916i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
229 $$1$$
good3 $$1 + (0.677 + 0.735i)T$$
5 $$1 + (0.789 + 0.614i)T$$
7 $$1 + (0.879 - 0.475i)T$$
11 $$1 + (0.401 + 0.915i)T$$
13 $$1 + (0.789 + 0.614i)T$$
17 $$1 + (0.789 + 0.614i)T$$
19 $$1 + (-0.789 + 0.614i)T$$
23 $$1 + (-0.546 - 0.837i)T$$
29 $$1 + (-0.879 + 0.475i)T$$
31 $$1 + (0.401 + 0.915i)T$$
37 $$1 + (0.945 + 0.324i)T$$
41 $$1 + (-0.0825 - 0.996i)T$$
43 $$1 + (-0.945 - 0.324i)T$$
47 $$1 + (0.0825 - 0.996i)T$$
53 $$1 + (-0.677 - 0.735i)T$$
59 $$1 + (-0.945 + 0.324i)T$$
61 $$1 + (-0.0825 + 0.996i)T$$
67 $$1 + (0.0825 - 0.996i)T$$
71 $$1 + (0.401 - 0.915i)T$$
73 $$1 + (0.245 + 0.969i)T$$
79 $$1 + (0.879 + 0.475i)T$$
83 $$1 + (-0.945 + 0.324i)T$$
89 $$1 + T$$
97 $$1 + (0.245 - 0.969i)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

−21.306585585132093711467639052450, −20.62940548859815372051020902554, −19.9875050576161440147234135713, −18.89955706022568471040059234310, −18.35217052977799426758857813523, −17.57068968678397834900295562581, −16.884765426394766219762286431278, −15.76832666288852773568628567931, −14.854507783060508463486347084402, −14.08070275718895316882318975719, −13.429184735997899580674783868783, −12.80669679837790951557902946757, −11.75617198975867502498584014989, −11.10437438216339096095094259675, −9.67945349525453058836055678574, −9.0670213631574911094595446808, −8.207419902073136190695837867397, −7.73744203148235052067582679141, −6.18332215354398514561791104316, −5.8552577824873228704599910541, −4.65171492752060179500162303479, −3.415938295876791958498568461326, −2.416717664229957435219197347026, −1.48332474990797938941585937566, −0.72408004191407666017983196914, 1.60378785307200015080985854051, 2.05457795604530888258245830089, 3.451311554743693489655435949070, 4.16452243114394651305649603849, 5.0844016212905639188258730151, 6.1854746746997285127279474102, 7.14176467299723990255860172559, 8.14392075122952920310240394335, 8.88209347070548509576819446782, 9.91986666385446605441339407098, 10.435975220977735015124012858048, 11.14018674127707820123591655137, 12.32835353628322928946269202509, 13.45139045754378647108541747676, 14.214680587773306758822588999720, 14.642684084642879045413243315378, 15.29238648585966970038543529592, 16.6492462272712934160663386309, 16.98466162849098351672000895054, 18.11806456469772524487896291415, 18.720623204135233188593502368446, 19.777703977776769909379684959565, 20.5983435598637918207612476094, 21.136246410730171861853882348100, 21.72079958130473272704586766371