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Properties

Label 1-1045-1045.389-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.790 - 0.611i$
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  + (0.615 + 0.788i)2-s + (0.719 − 0.694i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.913 − 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.0348 − 0.999i)9-s + (0.5 + 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.0348 − 0.999i)17-s + (0.809 − 0.587i)18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.374 + 0.927i)24-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (0.719 − 0.694i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.913 − 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.0348 − 0.999i)9-s + (0.5 + 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.0348 − 0.999i)17-s + (0.809 − 0.587i)18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.374 + 0.927i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.790 - 0.611i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.790 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.942988754 - 0.6639507584i\)
\(L(\frac12)\) \(\approx\) \(1.942988754 - 0.6639507584i\)
\(L(1)\) \(\approx\) \(1.532786492 + 0.05663095527i\)
\(L(1)\) \(\approx\) \(1.532786492 + 0.05663095527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
3 \( 1 + (0.719 - 0.694i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.882 - 0.469i)T \)
17 \( 1 + (-0.0348 - 0.999i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (-0.848 - 0.529i)T \)
59 \( 1 + (0.559 - 0.829i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.848 - 0.529i)T \)
73 \( 1 + (-0.438 - 0.898i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.615 + 0.788i)T \)
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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

−21.62109587125127423634087982584, −20.937120705172167377838658822421, −20.10332914452494691164601959475, −19.48876214089973717441736538167, −18.88785424430635131356655700731, −18.057263267574849198708819102203, −16.644572403024269662583277390282, −15.8587963737937144536004578626, −15.30618041101701851362371814264, −14.39283282874760513507394756163, −13.732614366784288665139297622674, −12.97658659578704828862472986855, −12.21215225351036035172657856292, −11.205384301436451096422913334082, −10.3162679321211932572197201842, −9.84015991310680739324914967269, −8.851769397385548300431929592380, −8.30307170421079026306833921197, −6.6101113550122580973701935915, −5.98508236207885939837480964745, −4.81743010494326518603286142124, −4.01522102362365263370868820125, −3.26964946476534930089792998186, −2.508636991853682849938734232316, −1.433651643471600783671613737940, 0.63969324451093461524935304678, 2.28379849608183942156647499752, 3.29028770642624689061332748711, 3.78306534378757301141725863489, 5.04859199734971065212662396455, 6.29716002919786909217071856219, 6.57740193483586047883759276154, 7.68285742641852492848746853116, 8.18619508625649099626744343584, 9.221833953138382042436963564172, 9.9458112660417277213664664029, 11.45457591978414702027303186088, 12.22942361104237980781826163113, 13.1546403243404187456426329320, 13.54465246792963832743649282645, 14.16680043780823417235334576005, 15.23313923710043927762772984740, 15.81607932342723709274834655735, 16.56873579924401231717269866990, 17.62696194996315963328838469513, 18.230154545438367977718237874488, 19.07536250442886176998620484925, 20.094309239522505524560531198773, 20.55681484856498112178425497811, 21.55141856906120500849671467206

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