Properties

Label 1-287-287.236-r1-0-0
Degree $1$
Conductor $287$
Sign $0.988 + 0.148i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (0.669 − 0.743i)2-s + (−0.5 + 0.866i)3-s + (−0.104 − 0.994i)4-s + (−0.913 − 0.406i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)10-s + (−0.913 + 0.406i)11-s + (0.913 + 0.406i)12-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.5 + 0.866i)3-s + (−0.104 − 0.994i)4-s + (−0.913 − 0.406i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)10-s + (−0.913 + 0.406i)11-s + (0.913 + 0.406i)12-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327129608 + 0.09939944074i\)
\(L(\frac12)\) \(\approx\) \(1.327129608 + 0.09939944074i\)
\(L(1)\) \(\approx\) \(0.9653128245 - 0.1928873421i\)
\(L(1)\) \(\approx\) \(0.9653128245 - 0.1928873421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−25.335123161240442284639778076289, −23.90819481685339388839323775737, −23.63994955153811786545218969844, −22.93742200822588549503231152996, −22.03613813461569832565293473495, −20.973092472193965227537715114306, −19.69159443693133371687738693167, −18.72019694457597942341956284686, −17.937169439471592986579258811468, −16.93132977046219903814233814244, −16.00912161107005284083898960094, −15.16947973251635139653784970788, −14.20003549702928880430261649928, −12.99104667692850193823342588834, −12.58621841990720289603556773851, −11.39309213104341299329309698353, −10.64042298329818460116386215303, −8.47553363402751505909616591396, −7.8306266293894787037961207573, −7.054476149538620286630748451949, −5.93109748304310340914306203517, −5.12299567614543117027802047751, −3.65321741303406633205452811046, −2.620990425475438741541937198217, −0.48437770464536670591273459889, 0.84634162199206274264192988460, 2.68426973065602677783149471144, 3.93135877582062801602026490968, 4.59526970896918887525342062033, 5.50486181906532397880293124594, 6.80226676908012727175634056687, 8.43159784672166477628494398330, 9.52001143622802986525018269063, 10.52012475753192577990053540897, 11.3046191261799776179003729442, 12.151393148534800900117097499564, 12.92039129032729249828037227573, 14.3584154121930955444673059120, 15.14597432440666233353075704647, 16.00737883733138454518895402040, 16.7811321387965860950503697084, 18.29299215309718503031391076733, 19.11853547245284915982396715641, 20.22286128567731524305661489579, 20.95738764810982743527672777666, 21.496930999970442142562264067246, 22.73949169146349376101731612464, 23.40806745868709889753305190122, 23.77017128377991618759873538312, 25.25878361999024343636201647945

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