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Properties

Label 1-2016-2016.221-r1-0-0
Degree $1$
Conductor $2016$
Sign $-0.984 - 0.175i$
Analytic cond. $216.649$
Root an. cond. $216.649$
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  + (0.707 − 0.707i)5-s + (−0.707 + 0.707i)11-s + (−0.258 + 0.965i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s − i·23-s − i·25-s + (0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.258 − 0.965i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (−0.5 − 0.866i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (−0.707 + 0.707i)11-s + (−0.258 + 0.965i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s − i·23-s − i·25-s + (0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.258 − 0.965i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (−0.5 − 0.866i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(216.649\)
Root analytic conductor: \(216.649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2016,\ (1:\ ),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05351532477 - 0.6044605653i\)
\(L(\frac12)\) \(\approx\) \(0.05351532477 - 0.6044605653i\)
\(L(1)\) \(\approx\) \(1.005606130 - 0.1406189379i\)
\(L(1)\) \(\approx\) \(1.005606130 - 0.1406189379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.104457316793194555072744173221, −19.279049866327745416318263103465, −18.58315572024369471683644709991, −17.87738217378682399137924051511, −17.39116831822017901467081979619, −16.508218764487465744220636148541, −15.5195942631933657173231172716, −15.12884383498355197266414309079, −14.12298940655939961811750919010, −13.513614393405831499388784356878, −12.96426498321015220493402119133, −11.92657379953154760675579306700, −11.08084527279990841328281915063, −10.377568526597234936827847511158, −9.90656231217256101779960575723, −8.891013887195236523810443064728, −8.02134745035405767452800163270, −7.32878801903140282415020722777, −6.367148330579801786489968221142, −5.635139904322917603288127435146, −5.0878895098287284516650359861, −3.65707097727056689850359640712, −3.05170338280695747587566998504, −2.18899189094750501862799337969, −1.15315910201252868366157625443, 0.10387994974313026249477106247, 1.20843076179790228630714856184, 2.172943329568731296538825048375, 2.836256601287064672083756686904, 4.2627152851909225555063509395, 4.882961748919941137866812654557, 5.489251413192940215154761338641, 6.65328902144072906215659677829, 7.17183457471489461073629891420, 8.28327410008013803954704364922, 9.00657241443538755665504778191, 9.70101207869521624387954745934, 10.29947491981491532722873034529, 11.34441024472224303866311883394, 12.15507488625748925155572511308, 12.74896248417713761205559060736, 13.63929853526277002432772561203, 14.08048063590374192926507929271, 15.01359843545348301778770519067, 16.05490399972852208862779804187, 16.33761556140797407979462952980, 17.27370908572716422596341393003, 18.1164170091101026159993575216, 18.32011599277202410646399023486, 19.664737814207052014649019415902

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