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Properties

Label 1-287-287.121-r0-0-0
Degree $1$
Conductor $287$
Sign $0.113 - 0.993i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.913 − 0.406i)2-s + (−0.866 − 0.5i)3-s + (0.669 + 0.743i)4-s + (0.978 − 0.207i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)9-s + (−0.978 − 0.207i)10-s + (0.207 − 0.978i)11-s + (−0.207 − 0.978i)12-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.104 − 0.994i)18-s + (0.994 + 0.104i)19-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.866 − 0.5i)3-s + (0.669 + 0.743i)4-s + (0.978 − 0.207i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)9-s + (−0.978 − 0.207i)10-s + (0.207 − 0.978i)11-s + (−0.207 − 0.978i)12-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.104 − 0.994i)18-s + (0.994 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.113 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5413241613 - 0.4831757851i\)
\(L(\frac12)\) \(\approx\) \(0.5413241613 - 0.4831757851i\)
\(L(1)\) \(\approx\) \(0.6189594767 - 0.2749590801i\)
\(L(1)\) \(\approx\) \(0.6189594767 - 0.2749590801i\)

Euler product

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

−25.86103133432963561202348412078, −24.92924552391527208677678917121, −24.15709648086745087542239691285, −22.96686299484517188344839315527, −22.25942408339083656270401821739, −21.09751115377506665121241263606, −20.42418932344507106301795685591, −19.10332636172095118018766122914, −17.9827669423357065239636184606, −17.63778564209861078549014568190, −16.70541175457419349160597437071, −15.94535855178820622293062059562, −14.85710444043733794718495994863, −14.02338016344095166312141024534, −12.40982740134660832327773336351, −11.431224404119023776529239310549, −10.46709460960708792723752607124, −9.55990360417814325505355169670, −9.16958463096117267507894166983, −7.197832682104997819154680642663, −6.71818831504044736096639229300, −5.488737383891410360121251313665, −4.70217478357927246621592212282, −2.60505970343876352625979029282, −1.26235739686394964566416348876, 0.85085588277113221242710348252, 1.908534143748405154277936933950, 3.25615286350975135728759097566, 5.18954626910361554027077478359, 6.09819103045152294343134233297, 7.11199198704871244711987852451, 8.23373520793234428575605925445, 9.32062916405266195689797219265, 10.39975905346762296952865649292, 11.00658468739647364522289026200, 12.20496021226673708022801405920, 12.90993485107374956813273836878, 13.91146328716860151472509962702, 15.565637750912268029221434325572, 16.648202402829293233372248249601, 17.25012098416670392559093279714, 17.903106063253079744750131198436, 18.80375182309435221841673513051, 19.65454876023686787237140251743, 20.76662367341763456643090725804, 21.86969225322389753827037852493, 22.15362799245930164182876413770, 23.7370481232817362638331989549, 24.71852018047085059865442340387, 25.12897336178304744075693444716

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