Properties

Degree $1$
Conductor $283$
Sign $-0.121 + 0.992i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.100 + 0.994i)2-s + (0.964 − 0.264i)3-s + (−0.979 + 0.199i)4-s + (0.860 + 0.509i)5-s + (0.359 + 0.933i)6-s + (0.480 + 0.876i)7-s + (−0.296 − 0.955i)8-s + (0.860 − 0.509i)9-s + (−0.420 + 0.907i)10-s + (−0.979 − 0.199i)11-s + (−0.892 + 0.451i)12-s + (0.359 + 0.933i)13-s + (−0.824 + 0.565i)14-s + (0.964 + 0.264i)15-s + (0.920 − 0.390i)16-s + (−0.824 − 0.565i)17-s + ⋯
L(s,χ)  = 1  + (0.100 + 0.994i)2-s + (0.964 − 0.264i)3-s + (−0.979 + 0.199i)4-s + (0.860 + 0.509i)5-s + (0.359 + 0.933i)6-s + (0.480 + 0.876i)7-s + (−0.296 − 0.955i)8-s + (0.860 − 0.509i)9-s + (−0.420 + 0.907i)10-s + (−0.979 − 0.199i)11-s + (−0.892 + 0.451i)12-s + (0.359 + 0.933i)13-s + (−0.824 + 0.565i)14-s + (0.964 + 0.264i)15-s + (0.920 − 0.390i)16-s + (−0.824 − 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.121 + 0.992i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.121 + 0.992i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.121 + 0.992i$
Motivic weight: \(0\)
Character: $\chi_{283} (60, \cdot )$
Sato-Tate group: $\mu(47)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.121 + 0.992i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.236928573 + 1.397348557i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.236928573 + 1.397348557i\)
\(L(\chi,1)\) \(\approx\) \(1.283707652 + 0.8445574315i\)
\(L(1,\chi)\) \(\approx\) \(1.283707652 + 0.8445574315i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.67528292301891885313619035211, −24.278323783027889476067637911934, −23.62494650033571086833844942043, −22.22458450367491403809672427920, −21.43452702828602292716425557137, −20.62467341733253788357139897254, −20.21728126832428449606366024195, −19.33455268198794425558411935777, −17.84032380850290997486699875191, −17.662488326773221935664012001493, −16.010690001742918528162641878237, −14.90897094590467929262162448357, −13.73709254691897258168783463697, −13.39698543569242061854267169068, −12.5669918304737899196229943483, −10.740591883034487684937289675357, −10.443022791167048841050555696686, −9.26418623034491209535997342762, −8.505176005410937370194727852701, −7.409420167997281531106844155575, −5.4164912169259378942531031483, −4.614779812463016257075562151585, −3.42940561600692919282454539858, −2.31293398333576224927811240594, −1.268051912076589214613101254602, 1.90625378095117955644752208690, 2.946445488169784904601016274133, 4.43641576708619803185020633130, 5.66462119402037448569336233503, 6.59374008550342744640367122365, 7.641228585305207905112902786421, 8.65439601277350918569953623861, 9.297732899296034990349419513544, 10.437110532275187696492701475971, 12.11408103664734457457667351332, 13.29694981584084017791372394948, 13.900815398908738242794128879140, 14.68628878225413566569945446836, 15.48745482485295681235547287373, 16.44286766399130215135888832820, 17.877842302015097300449688489148, 18.425479970201005092405355746640, 18.9294297744722413595823929055, 20.69545498618166893354547099172, 21.34133426723964650016715385955, 22.1678094651034922376193274023, 23.33964792734031823052017755562, 24.50958564883731148499742086020, 24.78338377375126913387514157376, 25.81736493819845074622878569733

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