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Properties

Label 1-1287-1287.355-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.352 + 0.935i$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)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 + (0.104 + 0.994i)19-s + (0.978 − 0.207i)20-s + (−0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (−0.669 + 0.743i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)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 + (0.104 + 0.994i)19-s + (0.978 − 0.207i)20-s + (−0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (−0.669 + 0.743i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7021384189 + 0.4859353216i\)
\(L(\frac12)\) \(\approx\) \(0.7021384189 + 0.4859353216i\)
\(L(1)\) \(\approx\) \(0.6649519660 + 0.3048941806i\)
\(L(1)\) \(\approx\) \(0.6649519660 + 0.3048941806i\)

Euler product

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

−20.66273365158599829127119483307, −20.126979391452808154089503236194, −19.412728547568614240975632886326, −18.72383461608339317113503221895, −17.96586826333748026354718313828, −17.232103619472342759652736036147, −16.33710761221329849026514476047, −15.5579347693975977253389273951, −14.79028359424994697449229139751, −13.57345597062702174428787121833, −12.93343234789637220955793556100, −12.18319861382542588652409424964, −11.61300897991700996705672140920, −10.91506628108104187921936394749, −9.86627442478092434094134505097, −8.85329141800843849104143628895, −8.66675634184285899645376471442, −7.73493604321974555694235778838, −6.56522195642715072996129219829, −5.22487308699731427447871380596, −4.653526827088991263883301211859, −3.73509602543435263623384038571, −2.67735680257531800739307441089, −1.87890145489582535316751038419, −0.628947788993661978220994499668, 0.68440479425449881012853340085, 2.07702435816003386846128190780, 3.668491726763799910530918976141, 4.090457377599007197686118461668, 5.17430897827797158843335910887, 6.29891993460039477118302536709, 6.884854988894196669508593189567, 7.78151907858940548909450798629, 8.10761008011560874723170927886, 9.412805725412994905505206761338, 10.11291693602923543490852204129, 10.91133673605286510767609586195, 11.64479665971587317245741820230, 12.91537806339864265960025175010, 13.76055735896085521773630903258, 14.30780549183770599965472446234, 15.1373261752589725079437068084, 15.79381379638396097702258375225, 16.50855712078656997244801776006, 17.34863414024161375093523503009, 17.99152315794450533874930845160, 18.75925852853145718404316244514, 19.536821426341232672337457031480, 20.084838894468979673078411686113, 21.18240586835808518461294642186

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