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Properties

Label 1-1045-1045.629-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.129 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069965780 - 0.9397742042i\)
\(L(\frac12)\) \(\approx\) \(1.069965780 - 0.9397742042i\)
\(L(1)\) \(\approx\) \(0.9751498582 - 0.4133921298i\)
\(L(1)\) \(\approx\) \(0.9751498582 - 0.4133921298i\)

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

−21.71320430484473397120831344250, −20.78464200735324060090903451239, −19.8435469053634599495684802072, −19.28679787124588620896164222900, −18.63802711151622110989957681872, −18.04026133560573392368907615357, −16.85119805741680646199637028365, −16.04031310358630023067498401824, −15.70595867051943294367769161068, −14.52342916905160276112556642769, −14.19210664003337848218685252213, −13.166635432605858099633981708860, −12.559487529421626386837026956260, −11.14343728001430131780815784905, −10.12371664288095629996449936953, −9.32476220807406196951311401554, −8.99111983429281761609171453081, −7.85501436763023610955071816516, −7.320459860225244961429610974940, −6.30244797074225843987630467105, −5.65495431165624387994762570697, −4.19116064378406869227301446321, −3.46318103682096730291155204073, −2.17145233206638532505851862332, −1.155133079884849788945715491703, 0.7373271029827353052337184616, 1.95541805943495735332202961534, 2.96053853035117546843015272353, 3.54115082949721918508029805673, 4.32390761471179626712384177278, 5.77653496193135355634916075566, 7.062610397606497482273417348542, 7.82480550650275964310475883450, 8.59321793874429792719994216416, 9.40868769166212817702555650335, 10.08766393033304036646579315152, 10.64408484875994521400381648451, 11.87168766610349933218849451645, 12.75685185503025732517381542910, 13.28931888596679528084093254094, 14.047258048690161675914431892643, 15.0741287303401532198000418769, 16.16096411318473439339501141871, 16.4762871001733413600736916296, 17.74715178803916586289163934271, 18.565850187401172486187991475352, 19.09438025962631797824319672445, 19.86158908252668401994602776360, 20.5084756048382059099055285041, 20.99265439497487509265039007919

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