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Properties

Label 1-2864-2864.237-r1-0-0
Degree $1$
Conductor $2864$
Sign $-0.679 + 0.733i$
Analytic cond. $307.779$
Root an. cond. $307.779$
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  + (−0.955 + 0.295i)3-s + (0.998 + 0.0529i)5-s + (0.427 − 0.904i)7-s + (0.825 − 0.564i)9-s + (0.175 − 0.984i)11-s + (−0.973 + 0.227i)13-s + (−0.969 + 0.244i)15-s + (0.990 − 0.140i)17-s + (−0.593 + 0.804i)19-s + (−0.140 + 0.990i)21-s + (0.0176 − 0.999i)23-s + (0.994 + 0.105i)25-s + (−0.621 + 0.783i)27-s + (0.648 + 0.760i)29-s + (−0.949 + 0.312i)31-s + ⋯
L(s)  = 1  + (−0.955 + 0.295i)3-s + (0.998 + 0.0529i)5-s + (0.427 − 0.904i)7-s + (0.825 − 0.564i)9-s + (0.175 − 0.984i)11-s + (−0.973 + 0.227i)13-s + (−0.969 + 0.244i)15-s + (0.990 − 0.140i)17-s + (−0.593 + 0.804i)19-s + (−0.140 + 0.990i)21-s + (0.0176 − 0.999i)23-s + (0.994 + 0.105i)25-s + (−0.621 + 0.783i)27-s + (0.648 + 0.760i)29-s + (−0.949 + 0.312i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2864\)    =    \(2^{4} \cdot 179\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(307.779\)
Root analytic conductor: \(307.779\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2864} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2864,\ (1:\ ),\ -0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2045807206 + 0.4680461534i\)
\(L(\frac12)\) \(\approx\) \(0.2045807206 + 0.4680461534i\)
\(L(1)\) \(\approx\) \(0.8888402268 + 0.01879482017i\)
\(L(1)\) \(\approx\) \(0.8888402268 + 0.01879482017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
179 \( 1 \)
good3 \( 1 + (-0.955 + 0.295i)T \)
5 \( 1 + (0.998 + 0.0529i)T \)
7 \( 1 + (0.427 - 0.904i)T \)
11 \( 1 + (0.175 - 0.984i)T \)
13 \( 1 + (-0.973 + 0.227i)T \)
17 \( 1 + (0.990 - 0.140i)T \)
19 \( 1 + (-0.593 + 0.804i)T \)
23 \( 1 + (0.0176 - 0.999i)T \)
29 \( 1 + (0.648 + 0.760i)T \)
31 \( 1 + (-0.949 + 0.312i)T \)
37 \( 1 + (-0.648 + 0.760i)T \)
41 \( 1 + (0.713 + 0.700i)T \)
43 \( 1 + (0.105 + 0.994i)T \)
47 \( 1 + (0.911 + 0.411i)T \)
53 \( 1 + (-0.621 - 0.783i)T \)
59 \( 1 + (-0.210 - 0.977i)T \)
61 \( 1 + (-0.749 + 0.662i)T \)
67 \( 1 + (-0.0352 - 0.999i)T \)
71 \( 1 + (0.938 - 0.345i)T \)
73 \( 1 + (-0.925 - 0.378i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (-0.278 + 0.960i)T \)
89 \( 1 + (-0.489 + 0.871i)T \)
97 \( 1 + (0.261 - 0.965i)T \)
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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

−18.634734496220703044606192652488, −17.77845497315915142233155661584, −17.347640780799971900703959668725, −17.072885201685245401394369415726, −15.9105681612042434468511759218, −15.24294653404175540781334635617, −14.538131694216229065436265661822, −13.73698786048285709468719359511, −12.79206766284359438652344995598, −12.36246859372350912002605584140, −11.8024908854817353341713138688, −10.84089906637455272524621430068, −10.15588204857063724407096364031, −9.50409032761635359393928281847, −8.806587292861200677374775480141, −7.53169479591792397360148460063, −7.16568849316010448792526463465, −6.06946735617234024129868830032, −5.54369700386464154387576551572, −5.014399826843612606940763906515, −4.17293099681670594892989889392, −2.63813208877966144404762166458, −2.0467108912731039742393451237, −1.31896731294538815701217114202, −0.09588442079054133344923350510, 0.99890257658723784412798749956, 1.56773739307611441854151169405, 2.84196424300608298114399345942, 3.78500943304131538907470698070, 4.73164409015609049279484740194, 5.24438277037425957712636155570, 6.14273270170904639748027284973, 6.650202185333035146077620304252, 7.529535881612549933031713722405, 8.45393624943181506795621407245, 9.48232733865668406059510704091, 10.06041738403018174313141373954, 10.70780154847033904588864249753, 11.161500981090260974430538065715, 12.29758005257442025239780194220, 12.64972897305679028372059548975, 13.72219411419720687672446887761, 14.38207746697784671410547629741, 14.74576223150853087477825167971, 16.13920950995176282849220513153, 16.70186707583679960071171722026, 16.93230566073066374960874246421, 17.73215852133991937549558404914, 18.39891142126602424172571714404, 19.06322383714804509087748180944

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