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Properties

Label 1-1045-1045.714-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.671 - 0.740i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s − 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s − 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (714, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683154316 - 0.7459487911i\)
\(L(\frac12)\) \(\approx\) \(1.683154316 - 0.7459487911i\)
\(L(1)\) \(\approx\) \(1.021018793 - 0.1237559910i\)
\(L(1)\) \(\approx\) \(1.021018793 - 0.1237559910i\)

Euler product

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

−21.38666033666263503957798671777, −20.39782120603587877776152028605, −19.654199792217124199130657834053, −19.070443674893349336931746880112, −18.144465238458843429634315729, −17.44460089279773315727323830434, −17.27075175521229438922448829916, −15.79677941111909635424764999048, −15.08280976241109328679434118078, −14.51049944723970106078139191856, −13.73836454667172972291900011334, −12.97030570787414174421432161555, −12.013730490114433745062907242778, −10.94048946817746750141190700175, −10.12707282014169923434505810859, −8.92663459826806382260507073216, −8.447712932538936324283480306852, −7.62422030889115052665804303588, −7.08236316250512524347120315308, −6.016915342529570739303036987114, −5.2479607395935854539012289510, −4.186509541599410961873897579971, −2.78123581264116427971259559386, −1.63261319629115868990007552157, −0.90279610899306054998780803182, 0.51046714784735129021690858394, 1.96805691249717870156227459165, 2.54566388108422918822782025468, 3.67569039619099602586555366985, 4.66002282041964204659830435138, 4.96483448274794550745449058423, 6.787848620503677847047571651480, 7.85096883971538425894484171247, 8.54679306186161812838328258968, 9.23256168631539221789358916084, 10.02373452219493044908989797620, 10.80706169767917111261127325607, 11.536027528802862302546649087932, 12.16471765890245704848418957958, 13.55598238256832828917648666498, 13.996697756447039261526243111543, 14.89746009208972974567221050336, 15.82720747078630674191183070650, 16.7576581661563126714079452339, 17.316557844708269263171066498605, 18.257325452294582174215581539032, 19.07552038257127785766201025186, 19.77656297204782332822346852691, 20.680692214478425208207111193887, 20.98013782760168298236477550897

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