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Properties

Label 1-283-283.61-r0-0-0
Degree $1$
Conductor $283$
Sign $0.611 + 0.791i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (0.991 − 0.133i)2-s + (−0.166 + 0.986i)3-s + (0.964 − 0.264i)4-s + (−0.944 + 0.328i)5-s + (−0.0334 + 0.999i)6-s + (0.784 − 0.619i)7-s + (0.920 − 0.390i)8-s + (−0.944 − 0.328i)9-s + (−0.892 + 0.451i)10-s + (0.964 + 0.264i)11-s + (0.100 + 0.994i)12-s + (−0.0334 + 0.999i)13-s + (0.695 − 0.718i)14-s + (−0.166 − 0.986i)15-s + (0.860 − 0.509i)16-s + (0.695 + 0.718i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.133i)2-s + (−0.166 + 0.986i)3-s + (0.964 − 0.264i)4-s + (−0.944 + 0.328i)5-s + (−0.0334 + 0.999i)6-s + (0.784 − 0.619i)7-s + (0.920 − 0.390i)8-s + (−0.944 − 0.328i)9-s + (−0.892 + 0.451i)10-s + (0.964 + 0.264i)11-s + (0.100 + 0.994i)12-s + (−0.0334 + 0.999i)13-s + (0.695 − 0.718i)14-s + (−0.166 − 0.986i)15-s + (0.860 − 0.509i)16-s + (0.695 + 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.859285540 + 0.9129446923i\)
\(L(\frac12)\) \(\approx\) \(1.859285540 + 0.9129446923i\)
\(L(1)\) \(\approx\) \(1.649766562 + 0.4559904245i\)
\(L(1)\) \(\approx\) \(1.649766562 + 0.4559904245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.991 - 0.133i)T \)
3 \( 1 + (-0.166 + 0.986i)T \)
5 \( 1 + (-0.944 + 0.328i)T \)
7 \( 1 + (0.784 - 0.619i)T \)
11 \( 1 + (0.964 + 0.264i)T \)
13 \( 1 + (-0.0334 + 0.999i)T \)
17 \( 1 + (0.695 + 0.718i)T \)
19 \( 1 + (-0.0334 + 0.999i)T \)
23 \( 1 + (-0.538 - 0.842i)T \)
29 \( 1 + (-0.979 + 0.199i)T \)
31 \( 1 + (-0.824 + 0.565i)T \)
37 \( 1 + (-0.420 + 0.907i)T \)
41 \( 1 + (0.920 + 0.390i)T \)
43 \( 1 + (-0.645 - 0.763i)T \)
47 \( 1 + (-0.0334 - 0.999i)T \)
53 \( 1 + (0.860 + 0.509i)T \)
59 \( 1 + (0.231 - 0.972i)T \)
61 \( 1 + (-0.892 - 0.451i)T \)
67 \( 1 + (0.100 - 0.994i)T \)
71 \( 1 + (0.964 + 0.264i)T \)
73 \( 1 + (-0.824 - 0.565i)T \)
79 \( 1 + (-0.645 + 0.763i)T \)
83 \( 1 + (0.480 - 0.876i)T \)
89 \( 1 + (-0.645 - 0.763i)T \)
97 \( 1 + (-0.538 - 0.842i)T \)
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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

−24.91516394730324736234477626187, −24.44670039185548402133964732313, −23.76646897676388757195213456838, −22.82295521169256946675983443858, −22.17447963170472593494446071848, −20.91241299329272437044232903890, −19.95291527929892940197524557470, −19.35795193272833088985233888022, −18.109315200049849752520745086973, −17.12845302545121350750197226806, −16.10385033887506589621821996462, −15.059640196663319911741585288872, −14.341899047883419028433094756419, −13.221435193149882423042841748848, −12.36304800135015403516757207198, −11.59743504504022460999490936821, −11.136989249030651438982312070395, −8.946517093060152433731910001012, −7.79248730133123655252884770908, −7.30687099755688993857236111883, −5.86786570128502349396694367138, −5.15564451608428225615631865726, −3.80079027227286569439535946856, −2.60361716176174830792305842332, −1.216822381686064855877959561277, 1.719065018103525842516193828964, 3.58723846220666855853044043693, 3.99363103659724860680468177204, 4.89144211869886444418759127124, 6.222143173074189090627502755721, 7.30444003029794615497743009825, 8.48870156163914001630440566258, 10.04431413302860798730058344584, 10.87875666732682667546494045342, 11.67191044894906512518806943176, 12.327469462269138431481481234185, 14.177671314161056585571112838812, 14.503498098168070398038448546758, 15.27810296521629444451411028685, 16.59734614523842350953902745878, 16.794441796145104674615735594548, 18.63401631307911321810147155959, 19.81484907055884937034431956934, 20.386504134002260281223754739229, 21.31760059209380460737059132329, 22.12221305473370034603591148373, 22.99372761992116670778245246977, 23.60219636754165213948595563660, 24.4831764884129577258391468324, 25.80379929039251983012585060570

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