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Properties

Label 1-1045-1045.1022-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.965 + 0.261i$
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  + (−0.642 + 0.766i)2-s + (−0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.766 + 0.642i)9-s + (−0.866 + 0.5i)12-s + (−0.342 + 0.939i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (0.642 − 0.766i)17-s − i·18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (0.173 − 0.984i)24-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.342 − 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.766 + 0.642i)9-s + (−0.866 + 0.5i)12-s + (−0.342 + 0.939i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (0.642 − 0.766i)17-s − i·18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (0.173 − 0.984i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.965 + 0.261i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.965 + 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03672486399 + 0.2755823940i\)
\(L(\frac12)\) \(\approx\) \(0.03672486399 + 0.2755823940i\)
\(L(1)\) \(\approx\) \(0.5549469426 + 0.08306716309i\)
\(L(1)\) \(\approx\) \(0.5549469426 + 0.08306716309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.342 - 0.939i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.642 - 0.766i)T \)
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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

−20.67106923187596112287381705834, −20.49392028659191085280243140447, −19.395664046814090497106066588591, −18.92037996245473846075011480750, −17.69381395030919564951131358717, −17.01658084633681626487962925472, −16.66233581595633147541155067563, −15.60093070791193579767375643386, −14.96631314715279066565420364633, −13.64887718235094917527746087648, −12.88065143164437070781035822285, −12.0979369915907228916137229678, −11.19876822971156793674918800938, −10.37941722705808121147631214602, −9.96468884264522509923594192028, −9.21398335148628046119398591493, −8.252031191370244783777286643819, −7.32904231779083349568391524603, −6.20828249634426632103193333610, −5.17460740368548149235822279449, −4.05177240066744299478323761812, −3.42089857843611822579806392752, −2.60138361761184147362068457340, −0.99324426376353825604778544947, −0.10642462909118876607450020082, 0.953951594823816060610567738139, 2.01913321376939274880354927453, 3.07807309975167795081265015234, 4.76648336786419899713012806102, 5.57141510165684871532374366839, 6.40840801647137051035281119124, 7.051288026987741680034418271749, 7.69811695984402876768657126883, 8.90363711925246894332521774511, 9.30655983853658411822045677125, 10.4319994654091679786738167017, 11.35510438913713696477020590032, 12.23471879551786151032657086622, 13.02329062789048928326665189657, 13.97925084003776177186004154347, 14.55785679961709230723831796935, 15.685620945980580699207370480666, 16.43381395378121419531118807379, 16.940247233925497855102591844970, 17.87161924522746885047489551284, 18.61394645388674752967080407070, 19.11777830770852136196440887815, 19.6502108837053944001044184979, 20.7632692310106269966009824740, 22.0295649458425165442977557633

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