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Properties

Label 1-1045-1045.178-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.255 + 0.966i$
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  + (0.406 + 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.913 − 0.406i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s − i·12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.913 − 0.406i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s − i·12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7102536620 + 0.9227369417i\)
\(L(\frac12)\) \(\approx\) \(0.7102536620 + 0.9227369417i\)
\(L(1)\) \(\approx\) \(0.7192446297 + 0.5915472125i\)
\(L(1)\) \(\approx\) \(0.7192446297 + 0.5915472125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.994 - 0.104i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−21.479671274964385837996030358051, −20.47571660833410734099230283278, −19.70102989099529264490663523082, −18.98713695952994003620767978211, −18.44455375068725945096189340906, −17.64185351321205757815710360019, −16.66203903480356996823378829560, −15.98136214950662234050351714744, −14.83774083784701552640979584425, −13.78200989783540657959527634143, −13.24690396748229891666133065530, −12.570906215565654507510877909238, −11.90355595682711501160290958363, −10.9470333630405702086552791099, −10.438491011068850088326743676834, −9.5019823292604170308580098246, −8.50846588918228100451979690476, −7.31699286336233934077208020568, −6.37238789579074393873609124399, −5.705894527812736741189709272735, −4.82196735116519506305942120037, −3.62288729129602082293398384417, −2.93220199377667582894382782351, −1.57210544055757608821456231207, −0.80988617717813426561254109584, 0.75176722485578107090580361407, 2.90219626265702682888243707332, 3.68904898908664108120213329926, 4.50738453747590806227764736223, 5.57102124137070103416771809055, 6.086998539633243959917474284648, 6.80429375931246731999397173041, 7.89566606291683914866488665727, 9.02760282604812027779882299141, 9.519891029575613770498635569643, 10.54487563386579521652423627964, 11.54779333389141087399212743353, 12.43248950144389577050083263835, 13.013494679998397592886141251142, 14.026142714097040167806448139333, 14.94943728559969544508071384142, 15.61798289536368760481948807498, 16.32580206989794941633288171707, 16.71245888591867047744132244554, 17.66370652648137829818913258049, 18.468684150232292870180329725247, 19.166570916705956477639600058566, 20.69184253724164386391366011665, 21.10785045386938333677458794495, 22.09481870198138201335688068042

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