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Properties

Label 1-1287-1287.907-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.0236 + 0.999i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.913 + 0.406i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (−0.669 + 0.743i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.913 + 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.913 + 0.406i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (−0.669 + 0.743i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.913 + 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0236 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0236 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.0236 + 0.999i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.0236 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2301948048 + 0.2248080672i\)
\(L(\frac12)\) \(\approx\) \(0.2301948048 + 0.2248080672i\)
\(L(1)\) \(\approx\) \(0.9312275648 - 0.3498690244i\)
\(L(1)\) \(\approx\) \(0.9312275648 - 0.3498690244i\)

Euler product

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

−20.84500179761289872134980252650, −20.19557545251956384917623382715, −19.3877470401355710439943209960, −18.57786201922973349523743611293, −17.59435291047844493220235243404, −16.70089263451601211891895946772, −16.01516630615270796304572925289, −15.46438319884518178247150932630, −14.94974720955471591694905511943, −13.78834760507488055235618985412, −13.145582719426506358289311322087, −12.31300388236469245796904671097, −11.802779322279904817448801601965, −11.05660496283944537938396742803, −9.644698124721079424509874279928, −8.76061089082765717354940993714, −8.15157306240889345089857057062, −7.09967207485925986220268994041, −6.567480081667881202839162135733, −5.37513958749944682028462895964, −4.87027117281521887209332557592, −3.76006287083922016312989794961, −3.13359240292458461487772409393, −2.06337063052344776791265042324, −0.09385209107961804399183596132, 1.29627505877469804721540362246, 2.52622360265941436087547530312, 3.481583305439598537742944530395, 4.02514011290301978554604137537, 4.77980320916564548739088293373, 6.233043441615352344838012824469, 6.540826533459470678164532653282, 7.65357782864007613951332122287, 8.54250735188962146071319476998, 9.86979714411538384687602973006, 10.466349886905385917478940336809, 11.043244114216643742688792414615, 12.0653944472395431682315210355, 12.60490089044859463920941311164, 13.37235847684524711631602194864, 14.36277344892687159740568096457, 14.80264529240170261604875391324, 15.80729013497420523549505911322, 16.29462227439535322547309048411, 17.36123240716585814851306438646, 18.52913835673919002961011166462, 19.21476580651214905050017396205, 19.68631877846505550947978719155, 20.35744411010718802777404654908, 21.23315138543168955747334195984

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