Properties

Label 1-283-283.29-r0-0-0
Degree $1$
Conductor $283$
Sign $0.525 + 0.851i$
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  + (−0.166 − 0.986i)2-s + (−0.824 − 0.565i)3-s + (−0.944 + 0.328i)4-s + (0.359 − 0.933i)5-s + (−0.420 + 0.907i)6-s + (−0.741 − 0.670i)7-s + (0.480 + 0.876i)8-s + (0.359 + 0.933i)9-s + (−0.979 − 0.199i)10-s + (−0.944 − 0.328i)11-s + (0.964 + 0.264i)12-s + (−0.420 + 0.907i)13-s + (−0.538 + 0.842i)14-s + (−0.824 + 0.565i)15-s + (0.784 − 0.619i)16-s + (−0.538 − 0.842i)17-s + ⋯
L(s)  = 1  + (−0.166 − 0.986i)2-s + (−0.824 − 0.565i)3-s + (−0.944 + 0.328i)4-s + (0.359 − 0.933i)5-s + (−0.420 + 0.907i)6-s + (−0.741 − 0.670i)7-s + (0.480 + 0.876i)8-s + (0.359 + 0.933i)9-s + (−0.979 − 0.199i)10-s + (−0.944 − 0.328i)11-s + (0.964 + 0.264i)12-s + (−0.420 + 0.907i)13-s + (−0.538 + 0.842i)14-s + (−0.824 + 0.565i)15-s + (0.784 − 0.619i)16-s + (−0.538 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.525 + 0.851i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.525 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03219739260 + 0.01796622476i\)
\(L(\frac12)\) \(\approx\) \(-0.03219739260 + 0.01796622476i\)
\(L(1)\) \(\approx\) \(0.3380375420 - 0.3473561123i\)
\(L(1)\) \(\approx\) \(0.3380375420 - 0.3473561123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.166 - 0.986i)T \)
3 \( 1 + (-0.824 - 0.565i)T \)
5 \( 1 + (0.359 - 0.933i)T \)
7 \( 1 + (-0.741 - 0.670i)T \)
11 \( 1 + (-0.944 - 0.328i)T \)
13 \( 1 + (-0.420 + 0.907i)T \)
17 \( 1 + (-0.538 - 0.842i)T \)
19 \( 1 + (-0.420 + 0.907i)T \)
23 \( 1 + (-0.892 - 0.451i)T \)
29 \( 1 + (0.860 + 0.509i)T \)
31 \( 1 + (-0.0334 + 0.999i)T \)
37 \( 1 + (0.593 + 0.805i)T \)
41 \( 1 + (0.480 - 0.876i)T \)
43 \( 1 + (-0.296 + 0.955i)T \)
47 \( 1 + (-0.420 - 0.907i)T \)
53 \( 1 + (0.784 + 0.619i)T \)
59 \( 1 + (0.100 + 0.994i)T \)
61 \( 1 + (-0.979 + 0.199i)T \)
67 \( 1 + (0.964 - 0.264i)T \)
71 \( 1 + (-0.944 - 0.328i)T \)
73 \( 1 + (-0.0334 - 0.999i)T \)
79 \( 1 + (-0.296 - 0.955i)T \)
83 \( 1 + (0.231 - 0.972i)T \)
89 \( 1 + (-0.296 + 0.955i)T \)
97 \( 1 + (-0.892 - 0.451i)T \)
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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

−26.126455453444835700812669316, −25.76234589680797597256315873809, −24.54711306033595653192040140645, −23.46354165905462753486740093855, −22.79065041879827705773437638259, −21.991019850099999282801963178258, −21.519641205131618951249435026, −19.73636155759149632312060215692, −18.64472229074907544277969366842, −17.81625687751267779922157975654, −17.35593658488906328614075410132, −16.00454042562254814473807940160, −15.38690388450615051493948255007, −14.856865860926178047640767540178, −13.33704926998083942786002897315, −12.57305869038503779374172282729, −11.03280298784356180897524625806, −10.0894499392059030343296174010, −9.55385505348407275496289189055, −8.09990985935738763821116481918, −6.86688102041388378826809092585, −6.05013180941781654984259523911, −5.37162529162747891773305637496, −4.060782807915936769391033992419, −2.60157148252062350663851326702, 0.02999532845941924065632180627, 1.35213879832567656431733569773, 2.562648598902925615486204492316, 4.26015102932698039741398135699, 5.07620855225654226453655086695, 6.32079321650336550929735627290, 7.644338681680942530342667517107, 8.76314482981223138581014110160, 9.96477277226643318409644679042, 10.597063114346307205465259570916, 11.88617624921747572390688304532, 12.496398547985324925953357988326, 13.40069333417761915193734361301, 13.941085467174051300502131345803, 16.31130643474955153385209615612, 16.51669838072315229288653185590, 17.68450041887063269738035099380, 18.425431705443692951582731802866, 19.415859141291580955468693992157, 20.19099203759539264940770084352, 21.2328534958507112352851120072, 21.97283612025340007198806093307, 23.078392982484693613401914433084, 23.64072404500636919682141730266, 24.64323919478862661840284333502

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