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Properties

Label 1-1045-1045.223-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.971 - 0.238i$
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  + (0.927 − 0.374i)2-s + (−0.275 − 0.961i)3-s + (0.719 − 0.694i)4-s + (−0.615 − 0.788i)6-s + (−0.406 − 0.913i)7-s + (0.406 − 0.913i)8-s + (−0.848 + 0.529i)9-s + (−0.866 − 0.5i)12-s + (0.999 − 0.0348i)13-s + (−0.719 − 0.694i)14-s + (0.0348 − 0.999i)16-s + (0.529 − 0.848i)17-s + (−0.587 + 0.809i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (−0.990 − 0.139i)24-s + ⋯
L(s)  = 1  + (0.927 − 0.374i)2-s + (−0.275 − 0.961i)3-s + (0.719 − 0.694i)4-s + (−0.615 − 0.788i)6-s + (−0.406 − 0.913i)7-s + (0.406 − 0.913i)8-s + (−0.848 + 0.529i)9-s + (−0.866 − 0.5i)12-s + (0.999 − 0.0348i)13-s + (−0.719 − 0.694i)14-s + (0.0348 − 0.999i)16-s + (0.529 − 0.848i)17-s + (−0.587 + 0.809i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (−0.990 − 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.971 - 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2713368950 - 2.239062849i\)
\(L(\frac12)\) \(\approx\) \(0.2713368950 - 2.239062849i\)
\(L(1)\) \(\approx\) \(1.128034314 - 1.161764796i\)
\(L(1)\) \(\approx\) \(1.128034314 - 1.161764796i\)

Euler product

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

−21.83872343252966853450232109061, −21.435238541053645985273907932281, −20.64683804812557657144174936492, −19.89713691575276466687082442837, −18.76400301074479379616868019663, −17.84045920236731035669037776150, −16.70380475256016326414814331535, −16.42079634056431185201178479894, −15.39900113194221555588270133887, −15.0733521593942975020349404632, −14.23689509267533658292055396806, −13.167327454531122437338532592, −12.46627028848151464602732000018, −11.61054469250438896680597865077, −10.91524232786973307973840662210, −10.03003457362035375250157129298, −8.80225853736370137882540292731, −8.40033329821527437826771551577, −6.90206744091807868627625854317, −6.1288159350884035999101567190, −5.42221158683685418824914964049, −4.72686557037356126061723107762, −3.44355732572568376794207506044, −3.24346812824291480649015014729, −1.757540348187625468702577692882, 0.72095528480888095006564794099, 1.558643793811160204251669118952, 2.77590091009947687404195160265, 3.561772074499503904288108586999, 4.62414132941783839460942931215, 5.66104019630772269408638492591, 6.35218378191741198489474892824, 7.17873896680754740590754364930, 7.82486363218305919621006747175, 9.235616623138079751264196596970, 10.24977850300610192836799066071, 11.19199176471321628452873806614, 11.57177498609921059952957878808, 12.737647234509200354463062618990, 13.20794265645562628484589410423, 13.819371202541539067504809239891, 14.54886146554005357401894288599, 15.6617909647363848411065889283, 16.508942523601409580180803974168, 17.15166623839757173491050791100, 18.38824755241314301097098322041, 18.88208811479821313346880483592, 19.76085807936827256852692829370, 20.41706884732078453614629360545, 21.07096349780957509167700651322

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