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Properties

Label 1-1045-1045.962-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.814 + 0.580i$
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  + (−0.994 + 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s − i·12-s + (−0.406 + 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.406 + 0.913i)17-s + (0.951 + 0.309i)18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (0.104 + 0.994i)24-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s − i·12-s + (−0.406 + 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.406 + 0.913i)17-s + (0.951 + 0.309i)18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (0.104 + 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.814 + 0.580i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8665596035 + 0.2772780793i\)
\(L(\frac12)\) \(\approx\) \(0.8665596035 + 0.2772780793i\)
\(L(1)\) \(\approx\) \(0.7647009990 - 0.04474745276i\)
\(L(1)\) \(\approx\) \(0.7647009990 - 0.04474745276i\)

Euler product

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

−21.05092978035322648809720953003, −20.6611209649275324406011061805, −20.07204203533626993081957014363, −19.21177789764736348120805777291, −18.212923878267429449520260700350, −17.49129511286137466365902283879, −16.91659921322040875781414760419, −16.00636458412901585283391390734, −15.412775675667759494174791848632, −14.5616082893091995539226977147, −13.88561908742131611370681994930, −12.42300009556420796038905300817, −11.61380271534429922179557515989, −10.83976561129045262996586113667, −10.21061258464273464755086408909, −9.52184796496847969157917257680, −8.55169812639401198075423728479, −7.94880715073548724460435986840, −7.15791995489331785011799614275, −5.781402590353697721063301260966, −5.008580272478743759574206899120, −3.89079143409072801904405741775, −2.89514597309383990143186746827, −1.98880708039058667276847794707, −0.53973844412383690170074822892, 1.215220538794775786573244461, 1.895003328995293727743026202515, 2.69407327914829911928369842807, 4.12094914363283408182969820505, 5.63422315201615893115156681090, 6.1867922603105677801622543016, 7.40854231213833492527561058205, 7.71690399887174034535981747064, 8.68333713436361577953404407792, 9.26731832295548224226347002455, 10.41816265077056818155263684324, 11.37573844690098937265339032298, 11.91248122853226526788068882097, 12.67024782078631021868869898456, 13.951097615341817098859947187054, 14.55895741603634061734783427137, 15.29453881548067224093427221411, 16.40684381929286165422271350237, 17.28685009044625204541887253710, 17.69334184816322208737908318577, 18.55584525107379336787523212916, 19.16201179662601347833963146126, 19.72663355047458911514079753396, 20.76206584760888184193026372002, 21.23892861906816562175266748062

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