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Properties

Label 1-1287-1287.394-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.980 - 0.196i$
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  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.669 − 0.743i)19-s + (0.809 + 0.587i)20-s + 23-s + (−0.978 + 0.207i)25-s + (−0.913 − 0.406i)28-s + (−0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.669 − 0.743i)19-s + (0.809 + 0.587i)20-s + 23-s + (−0.978 + 0.207i)25-s + (−0.913 − 0.406i)28-s + (−0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.980 - 0.196i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.980 - 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007735937819 + 0.07780241637i\)
\(L(\frac12)\) \(\approx\) \(0.007735937819 + 0.07780241637i\)
\(L(1)\) \(\approx\) \(0.5490378558 + 0.1047014647i\)
\(L(1)\) \(\approx\) \(0.5490378558 + 0.1047014647i\)

Euler product

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

−20.57569887848457281301094408836, −19.77173398199940634386327239815, −18.96585738811252728369135284245, −18.581031361957426323991946745639, −17.329449279874440352223464677937, −17.074054593175066718488855032069, −16.142079322686575274284468817888, −15.50852926233860993627620083373, −14.6719919254761725783045585416, −13.26383744572025842288828370859, −12.50839627543148544426931007756, −12.3008695679565597288238651452, −11.11449466205624128700432604469, −10.44578302504617003820209630041, −9.20877417112230586798826453313, −9.09140603758898432822347299349, −8.20552507319321459843182352836, −7.36909913188444357149130283787, −6.15774412296085199694885477919, −5.5420969900838456817451302127, −4.23551954541445614376694677700, −3.34543610299113458214936864208, −2.12351892431094511940558718248, −1.53628190277604805391279643436, −0.041565272216044011844214764463, 1.296662564117143484707938447851, 2.500589908628867238427904500878, 3.28550312828967966593376145448, 4.56807941976598108163958866119, 5.678911847464006757192784547753, 6.693283341701728950778356312417, 7.10435707674283358077929892268, 7.77051030173755841914388426224, 9.00531509094095766429833745114, 9.594945664511713461182317506072, 10.58910422759937796730301453367, 10.91056472660597463204874171742, 11.76109888847895325963951802198, 13.13774268142265203230766219321, 13.87281300150942491754073843940, 14.68850733270358182143939565610, 15.32745935796514372373024889124, 16.175026521692567377825679385395, 16.985254360478755864548218704583, 17.54834355833963008350820305954, 18.39819836875312519415286278849, 18.99201021753327009285313030224, 19.71383487868295593099260748942, 20.42615180090177877425037970135, 21.245507415037289643097680317100

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