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Properties

Label 1-283-283.106-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.974 - 0.224i$
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  + (−0.944 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.974 - 0.224i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.974 - 0.224i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.944 - 0.328i)T \)
3 \( 1 + (0.359 - 0.933i)T \)
5 \( 1 + (-0.741 + 0.670i)T \)
7 \( 1 + (0.100 - 0.994i)T \)
11 \( 1 + (0.784 - 0.619i)T \)
13 \( 1 + (-0.645 + 0.763i)T \)
17 \( 1 + (-0.420 - 0.907i)T \)
19 \( 1 + (-0.645 + 0.763i)T \)
23 \( 1 + (0.593 - 0.805i)T \)
29 \( 1 + (0.480 - 0.876i)T \)
31 \( 1 + (-0.997 + 0.0667i)T \)
37 \( 1 + (-0.296 - 0.955i)T \)
41 \( 1 + (-0.538 + 0.842i)T \)
43 \( 1 + (-0.824 + 0.565i)T \)
47 \( 1 + (-0.645 - 0.763i)T \)
53 \( 1 + (0.231 - 0.972i)T \)
59 \( 1 + (-0.979 - 0.199i)T \)
61 \( 1 + (0.920 + 0.390i)T \)
67 \( 1 + (0.860 + 0.509i)T \)
71 \( 1 + (0.784 - 0.619i)T \)
73 \( 1 + (-0.997 - 0.0667i)T \)
79 \( 1 + (-0.824 - 0.565i)T \)
83 \( 1 + (-0.892 + 0.451i)T \)
89 \( 1 + (-0.824 + 0.565i)T \)
97 \( 1 + (0.593 - 0.805i)T \)
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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

−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