Properties

Label 1-1287-1287.994-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.775 - 0.631i$
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.207 + 0.978i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.207 + 0.978i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.775 - 0.631i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (994, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4403578180 - 1.238386919i\)
\(L(\frac12)\) \(\approx\) \(0.4403578180 - 1.238386919i\)
\(L(1)\) \(\approx\) \(1.531393837 - 0.1649277987i\)
\(L(1)\) \(\approx\) \(1.531393837 - 0.1649277987i\)

Euler product

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

−21.2936840493496030266353173440, −20.342060374843560649376011147491, −19.96718543068139578639707376870, −19.10739946502117320738705713851, −17.6814359813456689084348266911, −17.01069265556165782617711354933, −16.55840456134295250816836847873, −15.7547150650062272920175555442, −14.96410853850695414262139587832, −14.025189550528626073249806228075, −13.33060042130837095894114955996, −12.83781537905087106953660623059, −12.11380319241506074968540782380, −11.09026462972125003150437176102, −10.34243681991883918375826911881, −9.27886881897342303560266791954, −8.339087163685864178404557050932, −7.52560125353723904918598704684, −6.67847042438437040183903209895, −5.7843322821833279553085695516, −5.07856856625057203637662367116, −3.98297724367599160419364423359, −3.689510995140644773810509725917, −2.17731546640356691927078537370, −1.31121380352423211114309725967, 0.15646142659307765119950644329, 1.820614733315650297542614826902, 2.65348898004288736936395778213, 3.13631213060625237661493109925, 4.315699234627546904609397079804, 5.23110263718100080205890862764, 6.10804092163911602649862121871, 6.729917015481270567865216513918, 7.47200104088705140502430597653, 8.94649578256684588113548587110, 9.61186557701340817087283332708, 10.73262072027585715725166031360, 11.14298394064640612822380817312, 12.034703382506816203631831556092, 12.855675877106013163672559561022, 13.53511435217322579207425398471, 14.3723451471308285555532129371, 15.1276228112184975459958716664, 15.53882856502253245596817984171, 16.4651634838746652219192733039, 17.56557956360623092605053555376, 18.65986389300924139044389650974, 18.84036543675557158927583310454, 19.883123715951773437309584353215, 20.62178496076174936674566083351

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