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Properties

Label 1-1045-1045.148-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.160 - 0.987i$
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  + (0.0697 − 0.997i)2-s + (−0.788 + 0.615i)3-s + (−0.990 − 0.139i)4-s + (0.559 + 0.829i)6-s + (0.207 + 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.241 − 0.970i)9-s + (0.866 − 0.5i)12-s + (−0.275 − 0.961i)13-s + (0.990 − 0.139i)14-s + (0.961 + 0.275i)16-s + (−0.970 + 0.241i)17-s + (−0.951 − 0.309i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.438 − 0.898i)24-s + ⋯
L(s)  = 1  + (0.0697 − 0.997i)2-s + (−0.788 + 0.615i)3-s + (−0.990 − 0.139i)4-s + (0.559 + 0.829i)6-s + (0.207 + 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.241 − 0.970i)9-s + (0.866 − 0.5i)12-s + (−0.275 − 0.961i)13-s + (0.990 − 0.139i)14-s + (0.961 + 0.275i)16-s + (−0.970 + 0.241i)17-s + (−0.951 − 0.309i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.438 − 0.898i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.160 - 0.987i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.160 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4455954642 - 0.5239259216i\)
\(L(\frac12)\) \(\approx\) \(0.4455954642 - 0.5239259216i\)
\(L(1)\) \(\approx\) \(0.6600785829 - 0.2268749129i\)
\(L(1)\) \(\approx\) \(0.6600785829 - 0.2268749129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0697 - 0.997i)T \)
3 \( 1 + (-0.788 + 0.615i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (-0.275 - 0.961i)T \)
17 \( 1 + (-0.970 + 0.241i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.529 - 0.848i)T \)
53 \( 1 + (-0.694 + 0.719i)T \)
59 \( 1 + (0.848 - 0.529i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.719 - 0.694i)T \)
73 \( 1 + (-0.999 + 0.0348i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.0697 - 0.997i)T \)
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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

−22.20700888423862237911137629642, −21.2295320939870446869046613440, −20.0043456447254968349426735827, −19.24226068872956357039788213060, −18.281953900954111996408778939471, −17.75886521434895036840934066686, −17.036624523250189699287216040498, −16.35139341817133984779277827156, −15.83213917130764024119580089090, −14.40641812041359395302182460709, −14.06087465096474419032005698057, −13.11901688252609695575884172721, −12.50224658766854971774687673723, −11.399622743143982735789788800589, −10.66030130197074575526441458028, −9.62043679138237327488961994524, −8.666544751407868967006429438392, −7.581440140237219502024036812605, −7.113570293054290360113205619577, −6.395376982858599625524548306168, −5.47745802076509577528812603276, −4.53524549342945358319830099844, −3.930430514928651778645632139157, −2.13974632677814476553310247717, −0.89934439813708856785450371972, 0.41284593020351263413873205463, 1.89022095224722540788785333318, 2.80928693014497239056634024429, 3.92025236421157045068110945181, 4.676261020262957922888362174219, 5.6186610689435042185847497405, 6.10838215142508505341596999144, 7.755859438153206538977770867115, 8.72224595120617771016207760931, 9.54590566808331208425981853160, 10.15866177071446653681274675883, 11.21033661225790944284891879832, 11.50403202966426447677166479985, 12.56436460930834955217663105218, 12.9609928586426935877390685603, 14.27175233451643489483099341020, 15.13034777223895096259120098046, 15.67163848153533063280944653925, 16.84026873093646029595047446479, 17.72951120462962880680951493016, 18.08420426257478697666349252443, 18.97165656488702188562468375689, 20.0065431546569056825747946707, 20.58375070659691320412564098044, 21.4906892272772923944908700238

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