Properties

Label 1-1045-1045.439-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.988 - 0.150i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s − 18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.766 + 0.642i)24-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s − 18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.766 + 0.642i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1571470463 + 2.072425370i\)
\(L(\frac12)\) \(\approx\) \(-0.1571470463 + 2.072425370i\)
\(L(1)\) \(\approx\) \(1.089794606 + 1.191603038i\)
\(L(1)\) \(\approx\) \(1.089794606 + 1.191603038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−20.993247761266794425072782836422, −20.23795131038007909751147342332, −19.686352174550626025119444537734, −19.20044418010412363879994239029, −18.02542222536796421022588672039, −17.35923719474229351098219971275, −16.33235907059642608376601972169, −15.42978300711945541617088643156, −14.61716804181424330068129762615, −13.7398618790128693520882418633, −13.23803025827910490821131984835, −12.643779229380669778441687811796, −11.80035714417900037267577134960, −10.90199427413757143757666033563, −10.18953266500258508900094216207, −9.086273291648155187153550423445, −7.76326937921541350436379523320, −7.248575520265545296929177252916, −6.26696054903013197988735295062, −5.69173304571075812620275055419, −4.41511539408639290044550901728, −3.513326136751329417557820536990, −2.67018603713886218138620302911, −1.68783447664415516268436588064, −0.56137647186598417617989308865, 2.240830076777773555269249617037, 2.71798471873622169564117620187, 3.986620893279031600327715725417, 4.43406007418464830652102272490, 5.52996678270203670710826580106, 6.15166523507626403455506230993, 7.127594812948090409529794975015, 8.319332046049521578295850170380, 9.064043478984891406173438304376, 9.86645817152669586701791197227, 11.05094700248280299702432439601, 11.630490324315427845765370842165, 12.47176092639575339995249655789, 13.45421019236148812009594993896, 14.17434049504172483220748612369, 15.04198958475151802074949152087, 15.48290204626681710355350083490, 16.39525408520922082064859020075, 16.72223824536871494212511269588, 17.93063730872812796361106752827, 19.00050653710159837552470008867, 19.95209017163685513314010695984, 20.56962767714864743629279003334, 21.42933805276810254170684394789, 22.06967571648152879830200890439

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