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Properties

Label 1-1287-1287.743-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.985 - 0.169i$
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  + (0.406 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1037886214 - 1.212877339i\)
\(L(\frac12)\) \(\approx\) \(0.1037886214 - 1.212877339i\)
\(L(1)\) \(\approx\) \(0.7118406979 - 0.5412217883i\)
\(L(1)\) \(\approx\) \(0.7118406979 - 0.5412217883i\)

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

−21.443914047241790746929796297525, −20.437756231564782587126371145874, −19.37619145068067483819123123475, −19.03608949390194329834640893456, −18.01626038147374884459052593179, −17.19373021518614320401102255361, −16.2985352393862463844672715500, −15.84688873369672844603169091280, −15.1248366212433022064403872482, −14.53024709099113465428584511996, −13.40709940413550718316018086004, −12.78679467286032247288183419696, −12.12360283653440319166520505648, −11.24490136675253161431510378127, −10.1470921361291999822695052765, −9.032506147314253067635269306486, −8.58949576356609690629693377749, −7.40536317107961458553931295161, −7.03884384920288156455645039548, −6.02316505790967172488784000462, −5.23086637884742615822250961585, −4.16617166465518700814424001034, −3.48148531314246468117948055908, −2.69667708014789338479859107535, −0.757081268031887228609087342312, 0.33828225326608250459879858452, 1.10133463165618091662206421507, 2.596351287547002925587043325230, 3.34662598008171550060924216036, 3.99955898504205338982281018344, 4.933777708358307303911906981260, 5.83622758215643133822382963380, 6.96177359519407523886756384493, 7.73875225869004673780107961022, 8.94339374192065766042790121657, 9.54672041047757123453872653819, 10.41105663700148475173869648051, 11.28685873601784668471805864805, 11.873656501035295667095475213220, 12.68444901990330541030235405001, 13.285527488937397034118521115079, 14.174616788058138999364124584478, 14.98968304586062394148849838692, 15.8504521135154004433847573126, 16.4600753602801574695522938154, 17.52933740001564096147032047438, 18.66871596396298792048914437151, 19.01873336296228098182934119187, 19.71042916353825431241274400165, 20.63391017546134952280814245747

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