Properties

Label 1-1287-1287.103-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.750 + 0.660i$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + 10-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.309 + 0.951i)28-s + (−0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + 10-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (0.978 + 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.309 + 0.951i)28-s + (−0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.750 + 0.660i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.750 + 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.576491446 + 1.350093499i\)
\(L(\frac12)\) \(\approx\) \(3.576491446 + 1.350093499i\)
\(L(1)\) \(\approx\) \(2.291487429 + 0.5208090766i\)
\(L(1)\) \(\approx\) \(2.291487429 + 0.5208090766i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 - T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−21.138756968444563610657968580897, −20.07319188074594803384667956582, −19.91418877215265974359940264694, −18.66209454828345831382608521063, −17.84258493337689059555057464731, −16.91883638600480329299396647712, −16.42740314810939210954888479081, −15.290105498793536322154205605018, −14.57268048465392234860003463328, −13.89195044565441910788232502442, −13.29024636809059257378482454459, −12.71160377326294007758083644806, −11.51878311868868479752808308728, −10.92061993586521137681513139854, −10.0627337293970515691338475594, −9.54850774555053990894696417628, −8.06554415267276256821087052097, −7.15448109155851155977240076743, −6.46450073732916327620017321271, −5.58283019208359890853949109530, −4.876049794796570459596532756659, −3.79583574733902716163146634331, −3.11333651894291971755728009895, −1.9186900971990627139219274814, −1.21703076005741532761339361657, 1.430844178444700578053832946874, 2.38865111983768324640421334756, 3.00720460802719070814071650210, 4.27971352246832346422589874476, 5.23049832691912956042484247517, 5.72309336421865209625937395388, 6.45799702068666875121302674135, 7.487763935558519179442943335540, 8.41670610599593842357888091503, 9.37350515641704663384564789958, 10.150486030165930694428456292752, 11.237379211428383286798774950694, 12.017177094884599275926024419889, 12.63671865042873669076294961939, 13.421965476414019055291212012988, 14.30327404822900018265983608267, 14.61816747126188555349331874478, 15.83862041346386420427548661465, 16.23540578473952353894646298157, 17.15510308895170730838057854185, 18.07537103818466063313083551166, 18.63804870645804038664556000435, 19.87334548987481338044220811056, 20.6473583932049593252046388318, 21.19232955798280478159066883496

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