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Properties

Label 1-1287-1287.4-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.307 + 0.951i$
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 + (0.978 − 0.207i)5-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)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 + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)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 + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (−0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3700142634 + 0.2693093834i\)
\(L(\frac12)\) \(\approx\) \(0.3700142634 + 0.2693093834i\)
\(L(1)\) \(\approx\) \(0.6978339598 - 0.2340220379i\)
\(L(1)\) \(\approx\) \(0.6978339598 - 0.2340220379i\)

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 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−20.95647130824702954331533672038, −19.94738873590080192724435195515, −18.92689430977297522110136185924, −18.559014064640987501509662409634, −17.7788156126656949318221530465, −16.82957710150923304339119349548, −16.5042672423547792308815857408, −15.49355476851364764525372295345, −14.79652526429499199952341831398, −14.058141355321695110493274470237, −13.21349885539477575947786651023, −12.744756778350837532909650382032, −11.45133516831441405996754268109, −10.1871976825698001237502740612, −9.8854306926301892021228530982, −9.027880435032792942268989486126, −8.327396519484123960052299691985, −7.105605219792868946905318516565, −6.549565336792868115436243936568, −5.77741585704327743338179690129, −5.12770135238104082597697117327, −3.9332475740247745570856958564, −2.75867675428207081325136996174, −1.72579475084965986731673677880, −0.193657520187816196396367760815, 1.383189549890361630980641103297, 2.00755518547376830493956219710, 3.259404014081267911952423010249, 3.79885920602061557967734894329, 5.0387314768785984986134275548, 5.87865706579635169360566885665, 6.86121013808334679561349029581, 7.929344002547174291304116482127, 8.90750649612407652217798196668, 9.5053222426439558299837743143, 10.37004328710281377889266786601, 10.64684295363743505563731591141, 11.99737933509583967106772768781, 12.674456313701253872610833084921, 13.24673668805354407536790191398, 13.95776264381490555454534339744, 14.78783103336891945992505257593, 16.13080353772473462517989196315, 16.887119117673586913272658914676, 17.3036937904132964799304174632, 18.27234080715332912848062090875, 18.92719035429085745969934765707, 19.723067578942775741385469560985, 20.316434996613730035024184778743, 21.236977800441478735348853252471

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