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Properties

Label 1-916-916.731-r1-0-0
Degree $1$
Conductor $916$
Sign $0.156 + 0.987i$
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  + (−0.789 − 0.614i)3-s + (−0.401 − 0.915i)5-s + (0.0825 + 0.996i)7-s + (0.245 + 0.969i)9-s + (−0.945 + 0.324i)11-s + (−0.401 − 0.915i)13-s + (−0.245 + 0.969i)15-s + (−0.401 − 0.915i)17-s + (0.401 − 0.915i)19-s + (0.546 − 0.837i)21-s + (0.986 + 0.164i)23-s + (−0.677 + 0.735i)25-s + (0.401 − 0.915i)27-s + (−0.0825 − 0.996i)29-s + (−0.945 + 0.324i)31-s + ⋯
L(s)  = 1  + (−0.789 − 0.614i)3-s + (−0.401 − 0.915i)5-s + (0.0825 + 0.996i)7-s + (0.245 + 0.969i)9-s + (−0.945 + 0.324i)11-s + (−0.401 − 0.915i)13-s + (−0.245 + 0.969i)15-s + (−0.401 − 0.915i)17-s + (0.401 − 0.915i)19-s + (0.546 − 0.837i)21-s + (0.986 + 0.164i)23-s + (−0.677 + 0.735i)25-s + (0.401 − 0.915i)27-s + (−0.0825 − 0.996i)29-s + (−0.945 + 0.324i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05890016435 + 0.05028771943i\)
\(L(\frac12)\) \(\approx\) \(0.05890016435 + 0.05028771943i\)
\(L(1)\) \(\approx\) \(0.5831453263 - 0.2176566854i\)
\(L(1)\) \(\approx\) \(0.5831453263 - 0.2176566854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.789 - 0.614i)T \)
5 \( 1 + (-0.401 - 0.915i)T \)
7 \( 1 + (0.0825 + 0.996i)T \)
11 \( 1 + (-0.945 + 0.324i)T \)
13 \( 1 + (-0.401 - 0.915i)T \)
17 \( 1 + (-0.401 - 0.915i)T \)
19 \( 1 + (0.401 - 0.915i)T \)
23 \( 1 + (0.986 + 0.164i)T \)
29 \( 1 + (-0.0825 - 0.996i)T \)
31 \( 1 + (-0.945 + 0.324i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (0.245 - 0.969i)T \)
43 \( 1 + (-0.546 + 0.837i)T \)
47 \( 1 + (-0.245 - 0.969i)T \)
53 \( 1 + (0.789 + 0.614i)T \)
59 \( 1 + (-0.546 - 0.837i)T \)
61 \( 1 + (0.245 + 0.969i)T \)
67 \( 1 + (-0.245 - 0.969i)T \)
71 \( 1 + (-0.945 - 0.324i)T \)
73 \( 1 + (-0.677 + 0.735i)T \)
79 \( 1 + (0.0825 - 0.996i)T \)
83 \( 1 + (-0.546 - 0.837i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.677 - 0.735i)T \)
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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

−21.67169037116964703993184543480, −20.85414677360159944966618002640, −20.030745065852599210675011275246, −18.99759517081330175782978901388, −18.35243834165196829190566970738, −17.50700985088135024908759062532, −16.58199797475216564321698532432, −16.20912452392569242964870793203, −15.012425191607733366012794628794, −14.6171377266512631179702620615, −13.51432264857276390002179731409, −12.57325118373092914251593453691, −11.51009919977077285147841487099, −10.9023933798875038004151619364, −10.37875109801580130201780238152, −9.5693982303820052849908251652, −8.27325682248741523634757109046, −7.260643179643185729356829525031, −6.655680214918668182360602166809, −5.65218884550735104178100167780, −4.59258771061258151755322722356, −3.8455988356267026071786962211, −2.98308108371290751548115109648, −1.440192389868552463593794828623, −0.02753804596104373814410181351, 0.70509037648429474135008108474, 2.05321971386504922832976408004, 2.92005363584019866808278214161, 4.65045497664347584112642417281, 5.20368738652607158146192132633, 5.781377032017012165912381266111, 7.18562326390656160986643332397, 7.716165451566125662307244170242, 8.723214507579857522140733663797, 9.544064485671546902585031252589, 10.74026630157383627943494141378, 11.56600866244209563722890227634, 12.19341464123835602534514734574, 13.02321639405274121958799909183, 13.353219641834333430509948903908, 14.94713648880634882851549329706, 15.688140061446378001201549454330, 16.21853745089148872199055963732, 17.29490512856213200080710090804, 17.91052019637240725673777358673, 18.54934371909516640975732522751, 19.493341231621215554486134628, 20.21473937247896984906706649252, 21.15122414591798363580378394934, 21.92804694654157083085775391501

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