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Properties

Label 1-1045-1045.124-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.984 - 0.174i$
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  + (0.0348 + 0.999i)2-s + (0.438 + 0.898i)3-s + (−0.997 + 0.0697i)4-s + (−0.882 + 0.469i)6-s + (0.104 − 0.994i)7-s + (−0.104 − 0.994i)8-s + (−0.615 + 0.788i)9-s + (−0.5 − 0.866i)12-s + (0.990 + 0.139i)13-s + (0.997 + 0.0697i)14-s + (0.990 − 0.139i)16-s + (0.615 + 0.788i)17-s + (−0.809 − 0.587i)18-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.848 − 0.529i)24-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (0.438 + 0.898i)3-s + (−0.997 + 0.0697i)4-s + (−0.882 + 0.469i)6-s + (0.104 − 0.994i)7-s + (−0.104 − 0.994i)8-s + (−0.615 + 0.788i)9-s + (−0.5 − 0.866i)12-s + (0.990 + 0.139i)13-s + (0.997 + 0.0697i)14-s + (0.990 − 0.139i)16-s + (0.615 + 0.788i)17-s + (−0.809 − 0.587i)18-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.848 − 0.529i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.984 - 0.174i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.984 - 0.174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119710345 - 0.09830414189i\)
\(L(\frac12)\) \(\approx\) \(1.119710345 - 0.09830414189i\)
\(L(1)\) \(\approx\) \(0.8443342990 + 0.5786710487i\)
\(L(1)\) \(\approx\) \(0.8443342990 + 0.5786710487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
3 \( 1 + (0.438 + 0.898i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.990 + 0.139i)T \)
17 \( 1 + (0.615 + 0.788i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.559 - 0.829i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.438 - 0.898i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.961 + 0.275i)T \)
53 \( 1 + (-0.374 - 0.927i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.374 - 0.927i)T \)
73 \( 1 + (0.719 - 0.694i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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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

−21.137334350920295354549996078785, −20.55910263322926615523422588081, −19.83578074930572692146786040014, −18.95352711891621447573174766936, −18.31110097281591414827348191862, −18.11648636963010320427280771650, −16.937157837104226847637091310636, −15.76891906586467607186020243000, −14.7486569354800868630918801368, −14.07996489448231179124160802568, −13.316100916986502465777676815895, −12.571847315921619296421322637811, −11.87961308090866158917128062503, −11.323934749806504485802401701667, −10.13543755809307623291568877149, −9.25124980401279723402845174744, −8.47676412846984896907482543749, −7.979175091148277672788910149278, −6.58829354357284145815839274428, −5.73794187634819369221166470864, −4.78770844216152186190699902780, −3.41220515617541571604884889186, −2.86410830490565276102632862158, −1.84072856195095120391715440785, −1.07839917040508883053931097789, 0.23182689016201319338184428029, 1.67244168609346701081616034103, 3.538642636179787917581508499196, 3.80424175752546289485690576813, 4.79547569852092938589819209914, 5.71914777062768995413191657284, 6.58283119791824396296035094393, 7.77054411500400831015629108525, 8.20002778019855901153874220381, 9.160690148698859527558254974646, 10.027724554089790886795505008, 10.58276730030889250653687551895, 11.705184416568713622344667463733, 13.05759766198731318023356561753, 13.783611652063062827091074953954, 14.22407285963736721770266669294, 15.220598776472729044399413994554, 15.75427937110837228463361910181, 16.611300554526787000206398554235, 17.10326000112951974781954566893, 17.95037919826929569324705669835, 19.07039894373585010302493192688, 19.68367779077876581861551085366, 20.86936331764336392490461925670, 21.1756477250587038195132308163

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