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Properties

Label 1-1045-1045.729-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.689 + 0.724i$
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  + (−0.669 − 0.743i)2-s + (0.104 + 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s − 12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.809 + 0.587i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.104 + 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)6-s + (0.809 + 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s − 12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.809 + 0.587i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.689 + 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186756253 + 0.5091625554i\)
\(L(\frac12)\) \(\approx\) \(1.186756253 + 0.5091625554i\)
\(L(1)\) \(\approx\) \(0.9297439725 + 0.1429115747i\)
\(L(1)\) \(\approx\) \(0.9297439725 + 0.1429115747i\)

Euler product

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

−21.11243243009670261054996009374, −20.54752089561340258296839353693, −19.57615836120078185230472967787, −18.99209801258998559688227033330, −18.04455188991290204870913609333, −17.84794284328466713497943323709, −16.73847409475091864668291063809, −16.31766650066538550014098198182, −14.98172890206786739177925217929, −14.413678730092428354080473269271, −13.73140561406422380948697854460, −12.97148739386214757752611369110, −11.71387443263370766198817110667, −11.078397409786862178868556430133, −10.17719307148591865416358014984, −9.067960341356363913487985328187, −8.236814456546809391945930482396, −7.76830549680477193593817055134, −6.8852706657269331772667914735, −6.15376573905386127490062633504, −5.26298162831874299246660324604, −4.17072613518183827429954084332, −2.69497310648756173106545499865, −1.45853589324024030975771915886, −0.87944051741504052367403902084, 1.12135382542304569251748296504, 2.24174564125019581260418820702, 3.277706375685954223045051326525, 3.948915044851001237754088021852, 5.04907982318869670393557120213, 5.83417405196024655981962974102, 7.36512732762781681814894562104, 8.34586890114875557657600510315, 8.752968128551545695684954362542, 9.70268158162545059084421048533, 10.403290209888585786243454180273, 11.299706199650863006032860080507, 11.65218016093599264798656366060, 12.77322390033648241364433626076, 13.74321376537821587904701230163, 14.676694114039492167195127334769, 15.48894264247250567719940563599, 16.25937559827896047891465091281, 17.06429158588203738902654604321, 17.76067272422745385895550686801, 18.61390113465001198900635234050, 19.28434761189531602493742736761, 20.4151864231779502592498854483, 20.74309235268719206772415865220, 21.473929805733031026403059880247

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