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Properties

Label 1-1287-1287.193-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.688 - 0.725i$
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  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.994 − 0.104i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)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 + (−0.743 − 0.669i)19-s + (0.587 + 0.809i)20-s − 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.994 − 0.104i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)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 + (−0.743 − 0.669i)19-s + (0.587 + 0.809i)20-s − 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.688 - 0.725i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.688 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288614934 - 0.5540044577i\)
\(L(\frac12)\) \(\approx\) \(1.288614934 - 0.5540044577i\)
\(L(1)\) \(\approx\) \(0.9837831957 - 0.4498873371i\)
\(L(1)\) \(\approx\) \(0.9837831957 - 0.4498873371i\)

Euler product

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

−21.11576217688864607103412478435, −20.50268035213816036444805627157, −19.59585902595129658484370480850, −18.54374339353775529617212468475, −18.041125822545385882498358321208, −17.14576334640779767208258946764, −16.33818191463342825857436258138, −15.79755054069079894243193215956, −14.851586039080862113733252484, −14.434739027518641782932163863857, −13.61905330372932598009085172536, −12.63576111727694595432363742946, −11.85022858611380424375355441215, −11.25128781890637590941160278265, −10.143258078649607068202691963907, −9.01232987800262016240323467235, −8.08804958743176054223214952077, −7.77233399924575167881786926595, −6.87853643237632121452006163189, −5.96847714836484175463053839586, −4.867324671627957275151726905458, −4.31952705377183739065204974606, −3.537308323471711929130043781648, −2.33252541299550975431807364805, −0.66004492420907568184602644826, 0.88713407806200971701688385426, 1.97566098715620403353956572570, 2.86178791344793795114630229113, 4.10250757596468564196526844803, 4.38904059281785223807149964814, 5.43431381576857114812893367292, 6.37082682502635996863858203658, 7.66240439373219845041901027623, 8.44611231087324028897301049048, 9.035985753211709113910870594815, 10.28610143112953663317125163047, 11.00958623232022850714334162206, 11.50275906536702274363105381304, 12.44797259851423802978222510522, 12.80079248434180950566867875796, 14.11888526775462087210120716514, 14.58155437597468600376533422103, 15.39742857771444346645106043862, 16.084513429113084483758433898060, 17.48510065297372585437254604314, 17.87015902707875906372668986187, 18.95813731852681157522129920769, 19.468562736937922054751402955618, 20.11029421293852473841165085269, 20.95082452221208421217379332123

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