Properties

Degree $1$
Conductor $283$
Sign $-0.974 - 0.224i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.944 − 0.328i)2-s + (0.359 − 0.933i)3-s + (0.784 + 0.619i)4-s + (−0.741 + 0.670i)5-s + (−0.645 + 0.763i)6-s + (0.100 − 0.994i)7-s + (−0.538 − 0.842i)8-s + (−0.741 − 0.670i)9-s + (0.920 − 0.390i)10-s + (0.784 − 0.619i)11-s + (0.860 − 0.509i)12-s + (−0.645 + 0.763i)13-s + (−0.420 + 0.907i)14-s + (0.359 + 0.933i)15-s + (0.231 + 0.972i)16-s + (−0.420 − 0.907i)17-s + ⋯
L(s,χ)  = 1  + (−0.944 − 0.328i)2-s + (0.359 − 0.933i)3-s + (0.784 + 0.619i)4-s + (−0.741 + 0.670i)5-s + (−0.645 + 0.763i)6-s + (0.100 − 0.994i)7-s + (−0.538 − 0.842i)8-s + (−0.741 − 0.670i)9-s + (0.920 − 0.390i)10-s + (0.784 − 0.619i)11-s + (0.860 − 0.509i)12-s + (−0.645 + 0.763i)13-s + (−0.420 + 0.907i)14-s + (0.359 + 0.933i)15-s + (0.231 + 0.972i)16-s + (−0.420 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.974 - 0.224i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.974 - 0.224i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.974 - 0.224i$
Motivic weight: \(0\)
Character: $\chi_{283} (106, \cdot )$
Sato-Tate group: $\mu(47)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.974 - 0.224i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.05718396086 - 0.5032044168i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.05718396086 - 0.5032044168i\)
\(L(\chi,1)\) \(\approx\) \(0.5014521620 - 0.3416121079i\)
\(L(1,\chi)\) \(\approx\) \(0.5014521620 - 0.3416121079i\)

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

−25.92177032082769972597496661659, −25.3097636294544991516246733602, −24.50241263173366952558320508569, −23.5063479808422954842307537271, −22.21829889518909588423483492518, −21.368964101145159629705195147035, −20.10937464291848503579003474899, −19.834572913918755339447280905, −18.88623946276175193399908193404, −17.415476789961935897483023904450, −16.95715458464596445255984910559, −15.6467375015669563692548443914, −15.30294892164879328976024254228, −14.5982410783814886614002893536, −12.73077225916588965580332863784, −11.71375844099789364155463570170, −10.79623292642229221707633227883, −9.627550517294258037175194159709, −8.844637239323919200385543544393, −8.29377447704763619192677665416, −7.04092051941444032504267168975, −5.53249589685541327648904503118, −4.67421761164130714579825508757, −3.1860536776205189604028859706, −1.80592897309953159704792606484, 0.42706629300108825382691417135, 1.86559415731192769137300060848, 3.08720570081003892480125722060, 4.08712934969781272769669797408, 6.57395771953248362325563763007, 6.96165743633306406709869900746, 7.92449970402586750357592500530, 8.794305021348357651713759206488, 10.00782315479100187601528929237, 11.24453636650293527527024530444, 11.70126318642428900147318909653, 12.85016868593885242097702311666, 14.115126254210661960280454484, 14.815995199399178208172459201391, 16.34992108894694566208180376655, 17.0061013086612963255158005281, 18.10294253937590198565221523087, 18.8928072970329309038660321767, 19.56695252144818275121886439761, 20.125409700905115798137607723696, 21.26262750778163743159968086571, 22.57998535106935822952340845616, 23.49421665653704799167709184522, 24.480993232810438029152738102801, 25.1587318763005758686018015600

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