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Properties

Label 1-1287-1287.1217-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.679 - 0.733i$
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  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (0.743 − 0.669i)5-s + (−0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s + 10-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)20-s + (−0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.951 − 0.309i)28-s + (0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (0.743 − 0.669i)5-s + (−0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s + 10-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)20-s + (−0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.951 − 0.309i)28-s + (0.913 + 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5424822238 + 1.242550161i\)
\(L(\frac12)\) \(\approx\) \(-0.5424822238 + 1.242550161i\)
\(L(1)\) \(\approx\) \(1.153080472 + 0.7948503516i\)
\(L(1)\) \(\approx\) \(1.153080472 + 0.7948503516i\)

Euler product

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

−20.43109994859374443329242528564, −19.65902551088087113985308724555, −19.06508843744335632436145917961, −18.04338657318769341409090053133, −17.56104376606887160712844613527, −16.3418128762680786025082936584, −15.67207490539051189353183863549, −14.51706705290173517361967168337, −14.1644125220679958949359712122, −13.336588307731090440749253750581, −12.8423456315671879768622608015, −11.71660233303674739701541087778, −10.93498657793789705903177268447, −10.29325518228188115219113428880, −9.71925317074119496903254643062, −8.76013348930898425770730739419, −7.21871898350205848034612868938, −6.65944087537883588027204388044, −5.92145755006371306379181915868, −4.80516298736698831775738516760, −4.11077245576881112640633892053, −2.957424732685967224051881153192, −2.4670769783858482697035380968, −1.241855000232261600774022576919, −0.18279608703471871723865453671, 1.64132904255630451570734436655, 2.484311737912068282109678103824, 3.567063563238227891563226343922, 4.52840588589393545455628147207, 5.40017194056777191963494586074, 6.09564185768030904085642110567, 6.57713836622058466317311634821, 8.0432429988580559873692310678, 8.47408783924998923277563235295, 9.40789089816186806274881617142, 10.21962141718195879324301378735, 11.58301341340931834882926102450, 12.21713594691671776835305599768, 13.052331196941222532319876904301, 13.37724536032348683065836621454, 14.53341786851297768935252907449, 15.052802857849836866501578757042, 16.01039608597368326406470950992, 16.51794143826731458387496560031, 17.40259143111523130070185945693, 17.94532772525243568019037089498, 18.976269532377730119925996115535, 19.93947442158492423048001479937, 20.8350128787591901176054600781, 21.538664548057202108195496042976

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