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Properties

Label 1-1045-1045.104-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.504 + 0.863i$
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  + (−0.559 + 0.829i)2-s + (−0.990 + 0.139i)3-s + (−0.374 − 0.927i)4-s + (0.438 − 0.898i)6-s + (0.978 − 0.207i)7-s + (0.978 + 0.207i)8-s + (0.961 − 0.275i)9-s + (0.5 + 0.866i)12-s + (0.719 + 0.694i)13-s + (−0.374 + 0.927i)14-s + (−0.719 + 0.694i)16-s + (−0.961 − 0.275i)17-s + (−0.309 + 0.951i)18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.997 − 0.0697i)24-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.990 + 0.139i)3-s + (−0.374 − 0.927i)4-s + (0.438 − 0.898i)6-s + (0.978 − 0.207i)7-s + (0.978 + 0.207i)8-s + (0.961 − 0.275i)9-s + (0.5 + 0.866i)12-s + (0.719 + 0.694i)13-s + (−0.374 + 0.927i)14-s + (−0.719 + 0.694i)16-s + (−0.961 − 0.275i)17-s + (−0.309 + 0.951i)18-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.997 − 0.0697i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.504 + 0.863i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.504 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3555333113 + 0.6193385556i\)
\(L(\frac12)\) \(\approx\) \(0.3555333113 + 0.6193385556i\)
\(L(1)\) \(\approx\) \(0.5704626407 + 0.2951559938i\)
\(L(1)\) \(\approx\) \(0.5704626407 + 0.2951559938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
3 \( 1 + (-0.990 + 0.139i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (-0.961 - 0.275i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.615 + 0.788i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.990 - 0.139i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (0.241 - 0.970i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (-0.997 + 0.0697i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (0.882 + 0.469i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.559 + 0.829i)T \)
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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

−21.272348325005139465849893739583, −20.56271059835078629985573744637, −19.74729102713229226676723695459, −18.6858642839169760805393436773, −18.15331159348346033574664639036, −17.58266202671411731345774492374, −16.95166172026809348451113357207, −15.98098565690770446555114113059, −15.214088104149385566039806680087, −13.87957855980372539521429243364, −13.10217800085260378944374064456, −12.3288997440149537057825680661, −11.546263209450150668954308803217, −10.95853670683677145626829980037, −10.38184715438801051624909057431, −9.35353973899689876095666332698, −8.2885481856026571857923464425, −7.76429951826738672251372821406, −6.57873141418375395657956937505, −5.616417352601327012875301541451, −4.57977066954388996840102480274, −3.91921930536963210800945807559, −2.419660901517812955233094900389, −1.61037776823092648150579963202, −0.5003534967348834999669249463, 1.09176947209629033395340407278, 1.88969731791091686588070097384, 3.970961418036858737354441858684, 4.71302010532188612166676416787, 5.46266775430710870747676902381, 6.398168416263066719391830234332, 7.038516464403716779971521476364, 7.983430165463434895160713776970, 8.861271757217424038151595779968, 9.73061138481441792176674211985, 10.716477747412474283662310807986, 11.207645331172466233845341645565, 12.031527818118543736067415186973, 13.38141250224050150123857697794, 13.9739645137880735255629896856, 15.015430138533148751083305160902, 15.72262661555701895517737130493, 16.39572886872391569891612232479, 17.162343077226050310061134400, 17.86627329161619107767813415696, 18.26412145992412331049741389449, 19.1801806485746326261299465744, 20.22209633779999976929595409876, 21.09435518304807144121451326435, 21.95862801813349240158006295787

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