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Properties

Label 1-1023-1023.428-r0-0-0
Degree $1$
Conductor $1023$
Sign $0.920 + 0.390i$
Analytic cond. $4.75079$
Root an. cond. $4.75079$
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  + 2-s + 4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1023 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1023 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1023\)    =    \(3 \cdot 11 \cdot 31\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(4.75079\)
Root analytic conductor: \(4.75079\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1023} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1023,\ (0:\ ),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.493503613 + 0.7102139690i\)
\(L(\frac12)\) \(\approx\) \(3.493503613 + 0.7102139690i\)
\(L(1)\) \(\approx\) \(2.271439859 + 0.2456678580i\)
\(L(1)\) \(\approx\) \(2.271439859 + 0.2456678580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
31 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
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

−21.34900824905122521383786872545, −21.042457708603285694827793981229, −20.2155289829189671255797994709, −19.58205879202777313759146461438, −18.141478922183979269489097898314, −17.813972254285247758030139148722, −16.480759539987660376818273421602, −16.00902279875113317216477071547, −15.24229511376858890178878346548, −14.28076979113325187808600961835, −13.65566915812204697382272859185, −12.81038475254201213833518826366, −12.131229792463329095963361168123, −11.51013890239127819199956676313, −10.42303418830649255697284580415, −9.53239709764098471986032443286, −8.412828981693706431894261793033, −7.82591528379710716103376421162, −6.458007246017823876933438822627, −5.68670993679526254263738915778, −5.127756067515043544313923989674, −4.27142657111154546308490786726, −3.08340446285648380124041491297, −2.15105508360883072064872245502, −1.22964016533319849898793340021, 1.46669186447891979351740685879, 2.25100430628562878207199239230, 3.39564048234795370521326643400, 4.1431631759599322252551690052, 5.0313452067460667756162812081, 6.17297898863216103022194397137, 6.73506131738548350709331923818, 7.48452533727735635056363522817, 8.57518688252720247742590476699, 9.95756301239060395073802520612, 10.63655283100905428163372950334, 11.2785070977481258593484735456, 12.0506279953114044201851092332, 13.26655137795636028842522440955, 13.867995875755162835026857887260, 14.23567166550476836044756163923, 15.240413398250789991599330653086, 15.903502595638063482149954292353, 16.99328351558885253020868082054, 17.57293074455835691202156810221, 18.57086133892563764750696014261, 19.544517799778252273739556303110, 20.20836504248497056517513772200, 21.124088278170229707967597002113, 21.69103511406104571306417616137

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