Properties

Label 1-1045-1045.204-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.744 - 0.667i$
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  + (0.241 − 0.970i)2-s + (0.0348 − 0.999i)3-s + (−0.882 − 0.469i)4-s + (−0.961 − 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.5 + 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (0.719 + 0.694i)24-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (0.0348 − 0.999i)3-s + (−0.882 − 0.469i)4-s + (−0.961 − 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.5 + 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (0.719 + 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142533251 - 0.4367651958i\)
\(L(\frac12)\) \(\approx\) \(1.142533251 - 0.4367651958i\)
\(L(1)\) \(\approx\) \(0.8661967196 - 0.5560909508i\)
\(L(1)\) \(\approx\) \(0.8661967196 - 0.5560909508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
3 \( 1 + (0.0348 - 0.999i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (-0.997 + 0.0697i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.438 + 0.898i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (0.719 - 0.694i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.438 + 0.898i)T \)
73 \( 1 + (-0.615 + 0.788i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.241 + 0.970i)T \)
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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

−22.02682920288653751427526940021, −20.82032956677895679816014596694, −20.381478973922716748770574910560, −19.39653051079704575765083685669, −18.0596971058649035571018876557, −17.541720128328678830256304163914, −16.80322800240076736254689421931, −16.16941985736837114592859795085, −15.218423441471391857997041546570, −14.80509624393074206427247612775, −13.979779725062693319220254389831, −13.259165717010647956347846113416, −12.180681593294180938707315895172, −11.12506907855566357210661000511, −10.30336643022021599246076328219, −9.55847407843812394744025760845, −8.49161456933294775176435353630, −8.00813582633574769822059057730, −6.93600209589482985316710166498, −6.01763501813268299178492698217, −4.90824210864778205923116466419, −4.572780985333892827147156020979, −3.62122167966476181439331424567, −2.54197971874105178762449632865, −0.53051318639021352598570360544, 1.12077773656275147617689857056, 2.1285877790946308035299241624, 2.54637131628894420799314707680, 3.89874695771730060823938676216, 4.93572702392164302765915991981, 5.7218183364531173407678004311, 6.71383017362682977083369755056, 7.78817801686366600249058543657, 8.75035042715207710375810278815, 9.219748749647253656781695110976, 10.48411354001537101293305984717, 11.47321080141206848964629800127, 11.84457130601360630305176911625, 12.59602692851929686608916661628, 13.45918080792456509124799765274, 14.155869118035231686499888898306, 14.7869384879416695130229471859, 15.83373943310910681760517928226, 17.33312169506435418487148275232, 17.71402486386048786113045531134, 18.46673009194126382109339188269, 19.27278822484724071215260933021, 19.7041908427424760162774499546, 20.61621455075474666305048494462, 21.59251934746074836789918785809

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