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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 $0$

Origins

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Normalization:  

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

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