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Properties

Label 1-2011-2011.72-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.995 + 0.0909i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (0.5 − 0.866i)2-s + (−0.669 + 0.743i)3-s + (−0.5 − 0.866i)4-s + (−0.978 + 0.207i)5-s + (0.309 + 0.951i)6-s + (−0.913 − 0.406i)7-s − 8-s + (−0.104 − 0.994i)9-s + (−0.309 + 0.951i)10-s + (0.104 + 0.994i)11-s + (0.978 + 0.207i)12-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.669 + 0.743i)3-s + (−0.5 − 0.866i)4-s + (−0.978 + 0.207i)5-s + (0.309 + 0.951i)6-s + (−0.913 − 0.406i)7-s − 8-s + (−0.104 − 0.994i)9-s + (−0.309 + 0.951i)10-s + (0.104 + 0.994i)11-s + (0.978 + 0.207i)12-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.995 + 0.0909i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.995 + 0.0909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002701298578 + 0.05929780139i\)
\(L(\frac12)\) \(\approx\) \(0.002701298578 + 0.05929780139i\)
\(L(1)\) \(\approx\) \(0.6175960849 - 0.1204075856i\)
\(L(1)\) \(\approx\) \(0.6175960849 - 0.1204075856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 - T \)
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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

−19.13971175711996919027839639143, −18.78098451407605008293020182765, −17.86851226912157348901015667535, −16.86194726501931181739500998378, −16.50636477589869520190902762806, −15.9431228384374725826703667428, −15.13275915741341900338245739084, −14.27230015359450902922160850682, −13.40618381037530792183252595958, −12.81088156416067170748883205859, −12.17569984757458921403084572325, −11.5714641563544800914063796143, −10.828537450734098617809263804539, −9.27253503913352220636526490704, −8.83260125833989459619459370508, −7.81052318247096727523641689431, −7.041613237148410376798365376501, −6.75689891614803674864442105152, −5.58646592198286689102340238260, −5.194067084261890029787209743697, −4.08962969481434964789772355803, −3.26839079059162718949124243080, −2.32402668825333830341696333277, −0.58086892205759399032304530468, −0.0203295623928956283586966327, 0.89976224360440002062710041183, 2.35425018583592788908886919462, 3.29496753472233045712140680266, 4.05743407603212061843111274458, 4.43180963989111992270803292398, 5.42930980614860739949523065558, 6.34742568900012709900249037347, 7.04859628701208249636569283343, 8.2285675077658756611895994559, 9.32005230293548478073910743842, 10.01804260013728971100977648367, 10.496120558546347484241473403832, 11.19888192874322143851063343679, 12.02529931769485911827089946996, 12.70715337303122704350573607323, 12.99525382230003889527404322395, 14.58773961693580686656010390386, 15.04949649421784207072621524564, 15.35233049062256490837820922705, 16.6625530768335129273347225737, 16.96305559982676756567389768987, 18.08719399790821374100621247436, 18.85941201665199020759840423604, 19.59611389734100592006151767988, 20.18302372959440481858960010793

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