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Properties

Label 1-1045-1045.588-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.610 - 0.791i$
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  + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)18-s − 21-s + i·23-s + (0.809 − 0.587i)24-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (0.587 + 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)18-s − 21-s + i·23-s + (0.809 − 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.610 - 0.791i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (588, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.610 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262499383 - 0.6203963551i\)
\(L(\frac12)\) \(\approx\) \(1.262499383 - 0.6203963551i\)
\(L(1)\) \(\approx\) \(0.9625823894 - 0.3864315841i\)
\(L(1)\) \(\approx\) \(0.9625823894 - 0.3864315841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−21.7386791197424531292912837425, −20.5500633866379326865600453887, −19.96991528778244498904009952701, −19.32879437621488156019914309086, −18.41708413818967942437935461611, −18.01703203493121677248408077488, −16.64126846390784296212553566774, −16.122445820404095533586350964887, −15.44257638856200892727489255066, −14.858304200936807249933519699011, −13.83161678496914377340009296442, −13.322314917560330903823414960113, −12.35092953854477254003442934691, −10.82101622236419894353831977806, −10.22701240553090071855866841872, −9.35637903814447658248797071325, −8.77195612464326811229603399633, −8.06164013205184721528262250642, −7.059809893825358728455363263901, −6.37094198815868929524912939602, −5.284916724780244197679189040939, −4.34396505361950331696668264773, −3.179495318275491277475404682448, −2.32613792058767638094942546601, −0.86292614662239673290079064300, 0.941439683673353501295147127090, 1.99111627252185532593878079365, 2.826381763223091609809585324005, 3.807387171370400095595855382525, 4.26701246851318647579769879395, 6.17739467146449727181750441987, 6.97607617100399082034150306916, 7.886390256580276658623525363304, 8.65330501589730490313404537730, 9.45325273519407113224227831063, 9.95861034146095022223584005579, 10.989148097726698502510021369179, 11.91714151802564972947193969052, 12.78863862479864332916982960376, 13.474134901523838035115273791482, 13.90151853934820428372113645876, 15.2806279226627029827311344180, 15.94190738197493980802748481187, 16.942904174315860828917880885701, 17.66642500114044062907438177378, 18.81959854056539475763966141921, 19.016288873345260635502294928, 19.87052898320042720664033929686, 20.36685179464568763604841420142, 21.35000856948764409274200726877

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