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Properties

Label 1-1045-1045.203-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.0414 + 0.999i$
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  + (−0.898 + 0.438i)2-s + (0.927 + 0.374i)3-s + (0.615 − 0.788i)4-s + (−0.997 + 0.0697i)6-s + (0.207 + 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.719 + 0.694i)9-s + (0.866 − 0.5i)12-s + (0.970 + 0.241i)13-s + (−0.615 − 0.788i)14-s + (−0.241 − 0.970i)16-s + (0.694 + 0.719i)17-s + (−0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.559 + 0.829i)24-s + ⋯
L(s)  = 1  + (−0.898 + 0.438i)2-s + (0.927 + 0.374i)3-s + (0.615 − 0.788i)4-s + (−0.997 + 0.0697i)6-s + (0.207 + 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.719 + 0.694i)9-s + (0.866 − 0.5i)12-s + (0.970 + 0.241i)13-s + (−0.615 − 0.788i)14-s + (−0.241 − 0.970i)16-s + (0.694 + 0.719i)17-s + (−0.951 − 0.309i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.559 + 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0414 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0414 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.0414 + 0.999i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.0414 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159189742 + 1.112075443i\)
\(L(\frac12)\) \(\approx\) \(1.159189742 + 1.112075443i\)
\(L(1)\) \(\approx\) \(1.005680819 + 0.4887000789i\)
\(L(1)\) \(\approx\) \(1.005680819 + 0.4887000789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.898 + 0.438i)T \)
3 \( 1 + (0.927 + 0.374i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (0.970 + 0.241i)T \)
17 \( 1 + (0.694 + 0.719i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.469 + 0.882i)T \)
53 \( 1 + (-0.275 - 0.961i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.961 - 0.275i)T \)
73 \( 1 + (0.529 + 0.848i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.898 + 0.438i)T \)
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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

−21.09367424637016585040604917349, −20.29942590710135950022320310527, −19.891188465463145002245463755357, −19.10160771957191545483369644716, −18.21430590039311086790410320090, −17.82548396064272631416356065655, −16.708164968960751427488758551283, −16.04315009433974571364295879099, −15.13114585914608496848091145653, −14.10259781253821859974227255675, −13.407525148627361288595350445962, −12.69447557676087210414479219456, −11.6590032157690651215338651436, −10.86318645451182494979726593408, −9.98353731541481895621139519352, −9.29871624865027391978932511058, −8.41143561172706502828116597231, −7.669488556559159181235147163636, −7.1493929256720540990481816194, −6.11090815130965283214447070505, −4.44792587638280795430957024611, −3.47579127546732958820401784190, −2.88477152217170710847217675757, −1.547122309345304519703037688038, −0.95050808094115870429865445979, 1.31616413641556707588071052226, 2.22143588875258509037691838215, 3.12454996545655719595763895694, 4.36172130365486734792706437311, 5.49014560431995350838699298478, 6.28673724197593383795762408496, 7.36999467457357318301670094969, 8.323464245287231794292518432147, 8.688443897662379152676524868809, 9.451258300817296896381593321909, 10.35913962830655041473813152954, 11.05554992617936898248399174935, 12.13471628917356545417121642183, 13.142032130692449597394057679, 14.32080504029900569008933582679, 14.7092955811416253219215027898, 15.59185324837519685673525317809, 16.10362342394442342774781184847, 16.95366465419506904807087935116, 18.04147303067156090066615933303, 18.74575069538513123695442047308, 19.12858658343103912810175961423, 20.131211973327699668397147849277, 20.836301545763060511342928409179, 21.423661694936771640702358134166

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