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Properties

Label 1-916-916.519-r1-0-0
Degree $1$
Conductor $916$
Sign $-0.409 + 0.912i$
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  + (−0.546 + 0.837i)3-s + (−0.986 + 0.164i)5-s + (−0.789 + 0.614i)7-s + (−0.401 − 0.915i)9-s + (0.879 + 0.475i)11-s + (−0.986 + 0.164i)13-s + (0.401 − 0.915i)15-s + (−0.986 + 0.164i)17-s + (0.986 + 0.164i)19-s + (−0.0825 − 0.996i)21-s + (−0.245 − 0.969i)23-s + (0.945 − 0.324i)25-s + (0.986 + 0.164i)27-s + (0.789 − 0.614i)29-s + (0.879 + 0.475i)31-s + ⋯
L(s)  = 1  + (−0.546 + 0.837i)3-s + (−0.986 + 0.164i)5-s + (−0.789 + 0.614i)7-s + (−0.401 − 0.915i)9-s + (0.879 + 0.475i)11-s + (−0.986 + 0.164i)13-s + (0.401 − 0.915i)15-s + (−0.986 + 0.164i)17-s + (0.986 + 0.164i)19-s + (−0.0825 − 0.996i)21-s + (−0.245 − 0.969i)23-s + (0.945 − 0.324i)25-s + (0.986 + 0.164i)27-s + (0.789 − 0.614i)29-s + (0.879 + 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ -0.409 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4687112386 + 0.7241080196i\)
\(L(\frac12)\) \(\approx\) \(0.4687112386 + 0.7241080196i\)
\(L(1)\) \(\approx\) \(0.6203639518 + 0.2566081542i\)
\(L(1)\) \(\approx\) \(0.6203639518 + 0.2566081542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.546 + 0.837i)T \)
5 \( 1 + (-0.986 + 0.164i)T \)
7 \( 1 + (-0.789 + 0.614i)T \)
11 \( 1 + (0.879 + 0.475i)T \)
13 \( 1 + (-0.986 + 0.164i)T \)
17 \( 1 + (-0.986 + 0.164i)T \)
19 \( 1 + (0.986 + 0.164i)T \)
23 \( 1 + (-0.245 - 0.969i)T \)
29 \( 1 + (0.789 - 0.614i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (-0.0825 - 0.996i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.0825 + 0.996i)T \)
47 \( 1 + (0.401 + 0.915i)T \)
53 \( 1 + (0.546 - 0.837i)T \)
59 \( 1 + (0.0825 - 0.996i)T \)
61 \( 1 + (-0.401 - 0.915i)T \)
67 \( 1 + (0.401 + 0.915i)T \)
71 \( 1 + (0.879 - 0.475i)T \)
73 \( 1 + (0.945 - 0.324i)T \)
79 \( 1 + (-0.789 - 0.614i)T \)
83 \( 1 + (0.0825 - 0.996i)T \)
89 \( 1 + T \)
97 \( 1 + (0.945 + 0.324i)T \)
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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

−21.803357065804829657824977338271, −20.08174383031356669745703887552, −19.879787113502946905399507439041, −19.21058645781166280256309278458, −18.39352403387884292698896999576, −17.285173976489758620278368885199, −16.86986730365647064826936679784, −15.96209364062329956895773584767, −15.25153306611667496365453228161, −13.85661920878106343406925843414, −13.53736771583782735311689969753, −12.29584856404894797232414977593, −11.95017323764511676375237543453, −11.1301490672006537762915453633, −10.16377793325502211368725356668, −9.07459123673150920435201661370, −8.11904812265368108856698034111, −7.07195459014493877937994772722, −6.896236657685760438603598677157, −5.64668370424311413951681769398, −4.630436679559076320598388264231, −3.64152390202423613151486402382, −2.641214593837707839437501218917, −1.133830627322926954789503888868, −0.37802964246651397259807238715, 0.63814382633373217419079467113, 2.48197799607084388645776549833, 3.42444453369600748875864320421, 4.35141849235698325153072847999, 4.96357094854760496748634839327, 6.34091081752738611506952682069, 6.76561488462542055509473486578, 8.04704853838245411132037915004, 9.095193835444412932747091274031, 9.69112755621146519800043228853, 10.56655656655254353142781322518, 11.65547670590519943512261912078, 12.01180847025177969508018275708, 12.7776735184006464661159910028, 14.31661014363437179709985113488, 14.884298020103146790572118894653, 15.81763213014049780960407218952, 16.08916063696210597815716456815, 17.12207854357833738525141185995, 17.86589607718638114831715590638, 18.89070770519549977929096454934, 19.79440776776707611018398562647, 20.11200337223350809083641306085, 21.385300852460590994740100200728, 22.13823806743435350350343160310

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