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Properties

Label 1-1287-1287.1102-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.139 + 0.990i$
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  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + 23-s + (0.669 + 0.743i)25-s + (−0.104 − 0.994i)28-s + (−0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.913 − 0.406i)5-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + 23-s + (0.669 + 0.743i)25-s + (−0.104 − 0.994i)28-s + (−0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6233430546 + 0.7171445521i\)
\(L(\frac12)\) \(\approx\) \(0.6233430546 + 0.7171445521i\)
\(L(1)\) \(\approx\) \(0.6444183508 + 0.3318567802i\)
\(L(1)\) \(\approx\) \(0.6444183508 + 0.3318567802i\)

Euler product

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

−20.53929391354166555806961980491, −19.64723524368832727892242267531, −19.44089583653838437381671417130, −18.491063167403629739277588393, −17.662850345296787028802423601398, −17.02808314453709825189433510175, −16.122997080442075723071486786223, −14.91815963756311352293789339202, −14.5516588448599760903985412917, −13.321153644691696555881788476518, −12.96746661455551480696257170730, −11.85669537527116468899027612880, −11.07325659638982289234959942945, −10.79839602145001162257713073146, −9.8530802673373304993551101455, −8.7878453976950295165074366514, −8.0981501876189007522619695715, −7.28199194875494677937756376331, −6.34295150382500193540814535094, −4.7821777772959639220577436811, −4.31118302123098339672666615263, −3.485100126205755641744083808583, −2.578549946903692477094370966645, −1.42355788490966444155399180712, −0.38380535457854887483061443568, 0.52592669528846075625103195903, 1.91545592080398554732808973863, 3.28434786012370890951117454046, 4.35641337972953529369192668624, 4.94972317620641224385741287692, 5.83692665573569942404383479109, 6.7974250367269496723435898095, 7.56452223843517672630673753681, 8.56209566464782962445674436231, 8.77831366388455071378177931426, 9.76909595187969442662628580915, 11.03624320441569025423755782253, 11.693920980511614300564099233773, 12.826556131654077796798750428913, 13.13616898333325928950125465470, 14.52556037663619962445746785972, 15.02277393429635577953084990436, 15.59695626143321083429185097473, 16.37232490975073722816764704029, 17.056540441439943961340081086, 17.929709295141341374792679961591, 18.737496758254662668137716109705, 19.19443193513218428562729915496, 20.185971168636634452625940795643, 21.09220825569731209118589501049

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