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Properties

Label 1-1045-1045.157-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.538 - 0.842i$
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  + (0.927 − 0.374i)2-s + (−0.275 − 0.961i)3-s + (0.719 − 0.694i)4-s + (−0.615 − 0.788i)6-s + (0.406 + 0.913i)7-s + (0.406 − 0.913i)8-s + (−0.848 + 0.529i)9-s + (−0.866 − 0.5i)12-s + (0.999 − 0.0348i)13-s + (0.719 + 0.694i)14-s + (0.0348 − 0.999i)16-s + (−0.529 + 0.848i)17-s + (−0.587 + 0.809i)18-s + (0.766 − 0.642i)21-s + (−0.984 − 0.173i)23-s + (−0.990 − 0.139i)24-s + ⋯
L(s)  = 1  + (0.927 − 0.374i)2-s + (−0.275 − 0.961i)3-s + (0.719 − 0.694i)4-s + (−0.615 − 0.788i)6-s + (0.406 + 0.913i)7-s + (0.406 − 0.913i)8-s + (−0.848 + 0.529i)9-s + (−0.866 − 0.5i)12-s + (0.999 − 0.0348i)13-s + (0.719 + 0.694i)14-s + (0.0348 − 0.999i)16-s + (−0.529 + 0.848i)17-s + (−0.587 + 0.809i)18-s + (0.766 − 0.642i)21-s + (−0.984 − 0.173i)23-s + (−0.990 − 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.538 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.524849359 - 1.930265727i\)
\(L(\frac12)\) \(\approx\) \(3.524849359 - 1.930265727i\)
\(L(1)\) \(\approx\) \(1.713748638 - 0.7659385575i\)
\(L(1)\) \(\approx\) \(1.713748638 - 0.7659385575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.927 - 0.374i)T \)
3 \( 1 + (-0.275 - 0.961i)T \)
7 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (0.999 - 0.0348i)T \)
17 \( 1 + (-0.529 + 0.848i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.0697 - 0.997i)T \)
53 \( 1 + (-0.469 + 0.882i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.882 + 0.469i)T \)
73 \( 1 + (0.829 - 0.559i)T \)
79 \( 1 + (0.615 - 0.788i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.927 - 0.374i)T \)
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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

−21.44747564255545456877878946636, −20.73720546651021483807731196322, −20.46241133794411467147670869744, −19.465389465211776612287740543964, −17.92405829991538624840045454380, −17.41521545237527804033429830702, −16.46202273335149097129093931508, −15.96521618430170167092974753189, −15.27689730975450388350016040502, −14.30339477397469779601064355930, −13.79640890751998197252668631365, −12.98902126091993168648972017386, −11.63489813697720643313799703508, −11.36168358198680526842076549697, −10.46394751428333348849335646519, −9.54129274014531551369279801143, −8.360344155935584851728280859656, −7.6418337348832034005118802006, −6.447385458324992692180547754365, −5.874673007778525535060121287028, −4.7017428808042672548888042942, −4.25777632267212039817944354100, −3.45143076831438253436763786539, −2.35046275403110393466444361983, −0.763896586541807510983125243542, 0.89321383113146425095579259376, 1.85610098449727684811807777100, 2.52619426234273698818839734057, 3.67989696238875985017579408660, 4.76968679910758426432611976628, 5.82271277563252789525629842414, 6.13424534261818174202988773402, 7.16233269250831023045284970734, 8.227606956323511891746652881555, 8.97570475903864069584872930108, 10.45823015466464321633226683119, 11.095128184037272732442678363140, 11.8791855570100498328313177481, 12.52884718937532409569103584257, 13.147141669906463030079956499814, 14.06398097923637775164066680501, 14.63387739293953124416769126971, 15.6838697472635574130005438396, 16.28121379802431233418101012944, 17.62236735248331728881518915930, 18.17072543815126792768962399819, 19.01250653079248586239857338163, 19.65191702252809255490655970786, 20.49363145568137937210357693327, 21.374815911351088955444065699298

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