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Properties

Label 1-1287-1287.41-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.0957 - 0.995i$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.743 + 0.669i)5-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (−0.406 + 0.913i)19-s + (−0.951 − 0.309i)20-s + 23-s + (0.104 + 0.994i)25-s + (0.207 − 0.978i)28-s + (−0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.743 + 0.669i)5-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (−0.406 + 0.913i)19-s + (−0.951 − 0.309i)20-s + 23-s + (0.104 + 0.994i)25-s + (0.207 − 0.978i)28-s + (−0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.0957 - 0.995i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.0957 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9590272580 + 1.055689753i\)
\(L(\frac12)\) \(\approx\) \(-0.9590272580 + 1.055689753i\)
\(L(1)\) \(\approx\) \(0.6037328989 + 0.8486116434i\)
\(L(1)\) \(\approx\) \(0.6037328989 + 0.8486116434i\)

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 \( 1 + (0.743 + 0.669i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.406 + 0.913i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.207 + 0.978i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.406 - 0.913i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.743 + 0.669i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−20.362018158749876427095981736840, −19.60194849229537813888597667778, −19.0317329644593213372594295201, −17.872006166344688360155304614833, −17.41732591429732729576147042795, −16.63111195164557001569848430698, −15.67045040594377722290582011114, −14.60932237376972685785601257574, −13.69255297532404616822260043274, −13.22077162532739802837766790426, −12.75570077469972017465797156462, −11.63665178520502047750117760907, −10.902561444424425136334164840522, −10.10146188405472520667916734914, −9.305222316477711300797465395098, −8.91440763046459969237890312176, −7.61593172149715667609975433573, −6.51466785623896017110941213594, −5.6391672337084948178042008954, −4.64430630178840936583524276853, −4.10757144497394443107168837761, −2.852847785658205106012317647049, −2.153319231231585180019279163253, −0.89903739644695423211179971912, −0.30626605677770892216755826941, 1.4831051702285556879982154426, 2.765078348507486216317579582009, 3.46640874339292245062716672211, 4.70018367446976270052522136166, 5.585428383666799886943183754070, 6.358655910622966997829224288853, 6.73469857712511583211844862175, 7.89819135662977765649147334048, 8.797156520783941943753992534214, 9.432068204290720350286661908506, 10.27033805360013381127709740529, 11.211002372471267483833864538354, 12.60901741496140530100898767801, 12.82350327667650280870241805399, 13.877021335404846085083607478951, 14.593374061083945473903257419686, 15.17766569025479109333137165705, 15.93028239629267437353308689054, 16.81738139061500288640095028984, 17.46175166239893534930859122316, 18.30695513350653550487056330973, 18.77717808871296755593658287574, 19.60923986898132439739056152800, 20.97836477894087776685982910346, 21.69019240375988199849232013998

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