Properties

Label 1-1045-1045.659-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.765 + 0.643i$
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  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s − 18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)24-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s − 18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4602140941 + 1.263145048i\)
\(L(\frac12)\) \(\approx\) \(0.4602140941 + 1.263145048i\)
\(L(1)\) \(\approx\) \(0.8143387633 + 0.6755281557i\)
\(L(1)\) \(\approx\) \(0.8143387633 + 0.6755281557i\)

Euler product

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

−21.080073542224047274761210216532, −20.39321074017960593174634622166, −19.4648211702509629140037165905, −19.1462845082024000214671929725, −18.451060766335551296188504136797, −17.579320661270563761871923249380, −16.90309791840180233928083113284, −15.43368623747713183491045972888, −14.86313200598322958132442204767, −13.96910211975949872777696577054, −12.933728173192492206127997694, −12.68002440682517743485385524692, −11.89866574272742390025584982103, −10.90264894192121302122111920042, −9.73523459259650378467856444552, −9.39264198718715636490490792496, −8.281528256756501349075662357128, −7.88308662382097531219120694463, −6.57043518518183471828867237667, −5.595290455970066393378143926755, −4.39491628604270650679482689951, −3.31873638112617246846681135662, −2.63905887239308377248359016472, −1.92132621479586860759536562405, −0.61848474966584737065245119634, 1.15603348049586003165963414472, 2.77248853414175173911598412400, 3.69946554221253419873561849046, 4.644435620927040729342334455043, 5.199396465633221897309662067973, 6.68213020113801852747173461560, 7.203225812126221575323001075211, 8.04043551549319975570081028851, 9.02598403630800222482790460483, 9.64557944761352105557201394011, 10.23043029961468850722454070409, 11.286750358850797975087349437817, 12.73399425236561014294392837969, 13.48931891190621143224622146206, 14.198299004222836605000745162280, 14.723080666008318902875531822196, 15.653083122287884647244638727222, 16.44721144206699151466213564215, 16.7427351355369849529170919396, 17.81705898784459735338349512102, 18.78163595296679718643099841509, 19.567386961998101262546387584721, 20.051294544178363497443241344471, 21.19309672640833720556862670594, 21.872364196265479594417698145559

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