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Properties

Label 1-1045-1045.192-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.941 - 0.338i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.275 − 0.961i)2-s + (−0.469 − 0.882i)3-s + (−0.848 + 0.529i)4-s + (−0.719 + 0.694i)6-s + (−0.743 + 0.669i)7-s + (0.743 + 0.669i)8-s + (−0.559 + 0.829i)9-s + (0.866 + 0.5i)12-s + (−0.898 + 0.438i)13-s + (0.848 + 0.529i)14-s + (0.438 − 0.898i)16-s + (−0.829 + 0.559i)17-s + (0.951 + 0.309i)18-s + (0.939 + 0.342i)21-s + (0.642 − 0.766i)23-s + (0.241 − 0.970i)24-s + ⋯
L(s)  = 1  + (−0.275 − 0.961i)2-s + (−0.469 − 0.882i)3-s + (−0.848 + 0.529i)4-s + (−0.719 + 0.694i)6-s + (−0.743 + 0.669i)7-s + (0.743 + 0.669i)8-s + (−0.559 + 0.829i)9-s + (0.866 + 0.5i)12-s + (−0.898 + 0.438i)13-s + (0.848 + 0.529i)14-s + (0.438 − 0.898i)16-s + (−0.829 + 0.559i)17-s + (0.951 + 0.309i)18-s + (0.939 + 0.342i)21-s + (0.642 − 0.766i)23-s + (0.241 − 0.970i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.941 - 0.338i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.941 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07545560294 - 0.4330495522i\)
\(L(\frac12)\) \(\approx\) \(0.07545560294 - 0.4330495522i\)
\(L(1)\) \(\approx\) \(0.4661351069 - 0.3180806280i\)
\(L(1)\) \(\approx\) \(0.4661351069 - 0.3180806280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.275 - 0.961i)T \)
3 \( 1 + (-0.469 - 0.882i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.898 + 0.438i)T \)
17 \( 1 + (-0.829 + 0.559i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.0348 + 0.999i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.788 - 0.615i)T \)
53 \( 1 + (0.0697 - 0.997i)T \)
59 \( 1 + (-0.615 + 0.788i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (0.997 - 0.0697i)T \)
73 \( 1 + (0.139 - 0.990i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.275 - 0.961i)T \)
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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

−22.22300374177510627093538447813, −21.28716329375483515845513158412, −20.18311953689561096441188924373, −19.59225017846752651060935524259, −18.625551622948558437365908816327, −17.51924346650186071807553650333, −17.16402316101207243375980065670, −16.46567063795571879951173952029, −15.54067890296550814105412075542, −15.26862507571061340091274237677, −14.16705351776374213327719444802, −13.418762705829686102691549099935, −12.50625228801604537356041948049, −11.29349470589302025413060183988, −10.48786974332177905255714799552, −9.57024678445783560775325238071, −9.37496142570073331508378743527, −8.0558212935313698141793463256, −7.15853569086725151808365882778, −6.39059460716315861505776602365, −5.52931609448608473371430099254, −4.6581444671529112808774365954, −3.96479546195607908382535973976, −2.80672282561146679977100183094, −0.821908381054195302297679440515, 0.3014063931224501144057675866, 1.73599151755546944721949387401, 2.41693418969317418175886153034, 3.34484356134532862136919525532, 4.66965101527809012344136657962, 5.46715470334875033177617809939, 6.6462802760834884346093002234, 7.28789373504378820068525555605, 8.60499664885129474744284728593, 8.966771196181053647662084771898, 10.190258116394391572869589762838, 10.853881603956294343895490575572, 11.82238606203765612395247598017, 12.45119478395347207325006152316, 12.873163513637319025275140912, 13.80801318689912066298127801465, 14.6595112336577180210839944593, 15.9379893859424564760039159582, 16.84694641953766333941478714717, 17.46012835001697210176136839913, 18.29763818677699307044834998513, 18.92037091644819705603919392183, 19.544463792461359411400482905833, 20.08302559809894718709541208997, 21.3424266382932706808952918441

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