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Properties

Label 1-1045-1045.819-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.978 + 0.207i$
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  + (0.961 − 0.275i)2-s + (−0.882 + 0.469i)3-s + (0.848 − 0.529i)4-s + (−0.719 + 0.694i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.559 − 0.829i)9-s + (−0.5 + 0.866i)12-s + (0.438 + 0.898i)13-s + (−0.848 − 0.529i)14-s + (0.438 − 0.898i)16-s + (−0.559 − 0.829i)17-s + (0.309 − 0.951i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.241 + 0.970i)24-s + ⋯
L(s)  = 1  + (0.961 − 0.275i)2-s + (−0.882 + 0.469i)3-s + (0.848 − 0.529i)4-s + (−0.719 + 0.694i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.559 − 0.829i)9-s + (−0.5 + 0.866i)12-s + (0.438 + 0.898i)13-s + (−0.848 − 0.529i)14-s + (0.438 − 0.898i)16-s + (−0.559 − 0.829i)17-s + (0.309 − 0.951i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.241 + 0.970i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.978 + 0.207i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.978 + 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07704560333 - 0.7361775451i\)
\(L(\frac12)\) \(\approx\) \(-0.07704560333 - 0.7361775451i\)
\(L(1)\) \(\approx\) \(1.141970904 - 0.3107026755i\)
\(L(1)\) \(\approx\) \(1.141970904 - 0.3107026755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.961 - 0.275i)T \)
3 \( 1 + (-0.882 + 0.469i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.438 + 0.898i)T \)
17 \( 1 + (-0.559 - 0.829i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (-0.997 - 0.0697i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.997 - 0.0697i)T \)
73 \( 1 + (-0.990 - 0.139i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−22.07166353424538424721212383057, −21.37831069610390973579681313612, −20.22535746340677339564411107635, −19.53228486040712881252383044714, −18.534593378331631981123576436289, −17.77588412904283530624034881644, −16.95042383590903962393247457257, −16.15773592682427220651668871408, −15.51310636468239411305019424822, −14.845669950446097950248910425943, −13.48288547382229386815935181984, −13.12048142530296197672987874437, −12.34186166281218661008574950584, −11.7010766458058723059772290137, −10.84463330476379051741292168890, −10.035493969732675615750832871327, −8.60620273727733157067626708848, −7.75154729530678955893385858174, −6.80613057784270137421759081019, −5.963536651725658530034138884642, −5.65612895675440030725672940424, −4.55692086749231257072616949222, −3.524988767769384642145346602165, −2.49352616367310631799276619484, −1.46483712241234149484017268549, 0.12097241226151119821485527370, 1.19446523526054329989794466979, 2.52215092867631309147452414009, 3.73710539560857245908974323034, 4.26856722171827114608441229579, 5.0840228606589867339530722256, 6.25994802629672425563825368156, 6.567013372401761623315244132348, 7.55774016692207491961829774735, 9.231202023509055941136076058877, 9.92273895260290043885239733507, 10.79708742053751482833689467262, 11.34124194909366945814849204220, 12.20670937636020577995864994753, 12.92024398052690717882634035123, 13.83067754825071921958371415745, 14.43754831121864196585971578603, 15.739033835946228051228816070639, 16.02303650101117519122681539980, 16.731023068013424629628696001743, 17.70828859684559106897181010526, 18.709805153543645133981229915822, 19.568510223956832673216194776, 20.43050171623182263227506179583, 21.046050469522668821642312529460

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