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Properties

Label 1-1287-1287.58-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.804 - 0.593i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.207 − 0.978i)20-s + (0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.743 − 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.207 − 0.978i)20-s + (0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.743 − 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.804 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1462808776 - 0.4445830178i\)
\(L(\frac12)\) \(\approx\) \(0.1462808776 - 0.4445830178i\)
\(L(1)\) \(\approx\) \(0.5038659782 - 0.1409778859i\)
\(L(1)\) \(\approx\) \(0.5038659782 - 0.1409778859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
7 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.207 + 0.978i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.994 + 0.104i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.207 + 0.978i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−20.89238799717291628058303207877, −20.105060364943143352353986203607, −19.21061142180972637348962311673, −18.984329464400686650252517182428, −18.30230271003460430082677244211, −17.14617345485331298844159021430, −16.67296404810455468118673586170, −15.7019082662779045131909836904, −15.28652585977193269082570850619, −14.51968264327020782225094682436, −13.481199335755566168008059684465, −12.33852505991037670863244605875, −11.71428532478671394234706598239, −10.741519515867131410341077224373, −10.21085687543210822687061094001, −9.29898981670709478997056075953, −8.56621019961265467069168206818, −7.51240450116852902358617589712, −7.0762773867074656068861175028, −6.190529752643583466839446834401, −5.37331684508544110912916413437, −3.855453750972698490079021967049, −3.08716022211060297456212129572, −2.13108165999536197922240546908, −0.72703269718998274547491234226, 0.2082320079354565632614017440, 1.02685258371053503971610406852, 2.25241160932328265133413122407, 3.37838425093092210254976151736, 3.949482198682301396048989213802, 5.26188449875692056900114693698, 6.43698297773052765858213253024, 7.107227400199498212904402774473, 8.10270283705076446181075938166, 8.65612543182333345471395268827, 9.508688363205535333953055243466, 10.26742953676040302167139584346, 11.037122805903066889561552076342, 12.03280817178700043591925364064, 12.57628587563280960274501720233, 13.15709260153431768517691073262, 14.56562145971271226074330988478, 15.49516316685309203105161107758, 16.08991610199935983517695669747, 16.830508133350268663985499079531, 17.21690925442374077870524150362, 18.50553886768226142523698343625, 19.17464218114323572199855405069, 19.46508618813926991446839379713, 20.4785118092734150525026598592

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