Properties

Label 1-1045-1045.14-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.700 - 0.713i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.374 − 0.927i)2-s + (0.961 − 0.275i)3-s + (−0.719 + 0.694i)4-s + (−0.615 − 0.788i)6-s + (−0.913 + 0.406i)7-s + (0.913 + 0.406i)8-s + (0.848 − 0.529i)9-s + (−0.5 + 0.866i)12-s + (0.0348 + 0.999i)13-s + (0.719 + 0.694i)14-s + (0.0348 − 0.999i)16-s + (−0.848 − 0.529i)17-s + (−0.809 − 0.587i)18-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (0.990 + 0.139i)24-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (0.961 − 0.275i)3-s + (−0.719 + 0.694i)4-s + (−0.615 − 0.788i)6-s + (−0.913 + 0.406i)7-s + (0.913 + 0.406i)8-s + (0.848 − 0.529i)9-s + (−0.5 + 0.866i)12-s + (0.0348 + 0.999i)13-s + (0.719 + 0.694i)14-s + (0.0348 − 0.999i)16-s + (−0.848 − 0.529i)17-s + (−0.809 − 0.587i)18-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (0.990 + 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5284228957 - 1.259015773i\)
\(L(\frac12)\) \(\approx\) \(0.5284228957 - 1.259015773i\)
\(L(1)\) \(\approx\) \(0.8654966423 - 0.4371673335i\)
\(L(1)\) \(\approx\) \(0.8654966423 - 0.4371673335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.374 - 0.927i)T \)
3 \( 1 + (0.961 - 0.275i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (-0.848 - 0.529i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.997 + 0.0697i)T \)
53 \( 1 + (-0.882 - 0.469i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.882 - 0.469i)T \)
73 \( 1 + (-0.559 - 0.829i)T \)
79 \( 1 + (0.615 - 0.788i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−21.88974475882042884825498242302, −20.54132396753381166339214534188, −19.98540571923608207149847082843, −19.26910652737588660656055194285, −18.61057397904844317402284356181, −17.64053499790422100962645005884, −16.849443889408255373795100752681, −15.932078029375727660433589418438, −15.506142364406596546989052097249, −14.72203506731064554902742893689, −13.87104700725560328565450349730, −13.20221318179147703151266937034, −12.56559687536110711661158586519, −10.74266342015416257657097381402, −10.212290970854836995710461855080, −9.47291084492019036898998135392, −8.577103418697939980834168282834, −8.034608601488574961100972884850, −7.00985700840917873417240891975, −6.389193943565301478851898276947, −5.219100349033491633555609016309, −4.22158346117028477420144452600, −3.42676598599059887103071527579, −2.255321769619825435526945164233, −0.86484992134829413480303792628, 0.33918619310803071095649552029, 1.73907297870997090113393605770, 2.35460081478775980216636370288, 3.388775486774750140909101625748, 3.95295805071172408584357123116, 5.17605813562181395297720456162, 6.67343750688535105401663085629, 7.312302977599125470423094265355, 8.45354321539315913953842575444, 9.16591615193789411725822759922, 9.534503182408054481353628759539, 10.53993129851295491558023906993, 11.6281722697148889452852680482, 12.30242589091158090992577801227, 13.1871559475537941663686308228, 13.65626488650553460864267125563, 14.53192585667375566767063961222, 15.65981706080020513569788437501, 16.30884821762298192607595660959, 17.426409950751544933304159569044, 18.30306356463744093301390592717, 18.93712062049158270062846489811, 19.45155822687054091380867950766, 20.20652482129693121434678854410, 20.80785930278334314124791488944

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