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Properties

Label 1-1287-1287.1136-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.414 - 0.909i$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.207 − 0.978i)5-s + (−0.994 − 0.104i)7-s + (0.587 + 0.809i)8-s − 10-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (0.587 + 0.809i)19-s + (0.207 + 0.978i)20-s + (−0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.951 − 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.207 − 0.978i)5-s + (−0.994 − 0.104i)7-s + (0.587 + 0.809i)8-s − 10-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (0.587 + 0.809i)19-s + (0.207 + 0.978i)20-s + (−0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.951 − 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.414 - 0.909i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.414 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5691842942 - 0.8851865892i\)
\(L(\frac12)\) \(\approx\) \(0.5691842942 - 0.8851865892i\)
\(L(1)\) \(\approx\) \(0.6947377818 - 0.4785654581i\)
\(L(1)\) \(\approx\) \(0.6947377818 - 0.4785654581i\)

Euler product

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

−21.60285471370993256973678558798, −20.3276026405042150989698421556, −19.42164110559914539093434631401, −18.805591425406568497622437657252, −18.1239768674283329654179703489, −17.5489808801246660594282914214, −16.44630717396573020708266004973, −15.95545583882456033332444153340, −15.246184627470086789944354499772, −14.31619018793337210869084836094, −13.84316476704507897585825943328, −12.99973360719844181376791328670, −12.04809779803043579177054926335, −10.89161317742323659335845720527, −10.070923260715756520107666607852, −9.49773558343589643085598579308, −8.66897217291764524994778358999, −7.522151789759607540806047143609, −6.92229098017542439388220834932, −6.29674184235622305627521771939, −5.47070858843895810723543518960, −4.44673027073004745146704837908, −3.300916217960932524542069677411, −2.573241007472314445618922608270, −0.86405488292902781726136714035, 0.631187709663807373717183534537, 1.6125142373928050038437958349, 2.62078132779458271354938642242, 3.75675270489398957052331302293, 4.23797433250323673468084726229, 5.51456727466455805251704104268, 6.10213250383420867209515847476, 7.675680064955408473633664515023, 8.22423046107010086619749174566, 9.382667852601813061080965612505, 9.65854286763089476102396240774, 10.480146534235098117783138520199, 11.57542408091332226531383119271, 12.305045136267814093579711822989, 12.87917015030801253751608133357, 13.53183322780498703432898490321, 14.298228895731543511431985484089, 15.604515854836464352977732475433, 16.358551402502007527874231962, 17.098432766712476896282889419119, 17.67093881097264203477916048599, 18.75688720539339269840154064472, 19.36463182165637937963604568402, 19.99632944209808800040775118982, 20.68359329014164604882929571072

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