Properties

Label 1-1287-1287.47-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.849 - 0.526i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.207 + 0.978i)5-s + (0.994 + 0.104i)7-s + (−0.587 − 0.809i)8-s − 10-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s + (−0.207 − 0.978i)20-s + (−0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (−0.951 + 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.207 + 0.978i)5-s + (0.994 + 0.104i)7-s + (−0.587 − 0.809i)8-s − 10-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s + (−0.207 − 0.978i)20-s + (−0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (−0.951 + 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.849 - 0.526i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.849 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2835146401 + 0.9955225879i\)
\(L(\frac12)\) \(\approx\) \(-0.2835146401 + 0.9955225879i\)
\(L(1)\) \(\approx\) \(0.6581968053 + 0.7297661817i\)
\(L(1)\) \(\approx\) \(0.6581968053 + 0.7297661817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
5 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.994 + 0.104i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + iT \)
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

−20.535663896286847146519956204650, −20.153978273678833153912017090552, −19.10970963489485978873280117605, −18.395448526070701556172794204428, −17.67301314798503600465639065566, −16.83566600090773081972415751281, −16.091227589818034052316819655740, −14.880643487107103505836652082824, −14.299028603906862393067976116614, −13.5051697485893625976633391484, −12.60900447255713557829891769329, −12.05784716238701318286802489054, −11.35092143255272067571202620172, −10.50426819530850131149836070731, −9.6924083012345088897328031342, −8.6696296546884486078805984750, −8.32020388678419057478964748336, −7.19889992442435505630727490411, −5.6674412249585027179954080386, −5.10914285308548007372951323523, −4.29919142298938336684895525562, −3.574141118004840098866389157735, −2.19645120009073329858782628652, −1.526843124814941480490686952606, −0.38157654140530098840178490906, 1.546867192329524938348643688170, 2.82213505686114112849347898305, 3.82732529344024087979757564881, 4.59125360101026359226531232015, 5.597396352278409691007140996624, 6.34928768520332188396556382345, 7.20075028633450528549734826476, 7.93587450074777621158883426604, 8.53674573757657847397735493203, 9.63091984382708574980079784852, 10.54697495940073696860803711459, 11.380919859280734093486037605982, 12.19817191164682442367630714543, 13.240082122556838379226869612587, 14.02411544170555660422280234623, 14.65944948434006728234319336082, 15.26162885563958567814112282514, 15.82798627922287077079838551533, 17.07384767938031478127981899874, 17.51976821365768148581294793444, 18.224829716783339791797198312589, 18.97989585802843698948241498222, 19.7257055997812371895973015768, 21.04198794635412971136629129049, 21.65773086680833752640951025214

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