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Properties

Label 1-1001-1001.307-r1-0-0
Degree $1$
Conductor $1001$
Sign $-0.289 - 0.957i$
Analytic cond. $107.572$
Root an. cond. $107.572$
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  − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s + 12-s + i·15-s + 16-s − 17-s − i·18-s + i·19-s + i·20-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s + 12-s + i·15-s + 16-s − 17-s − i·18-s + i·19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(107.572\)
Root analytic conductor: \(107.572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (1:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608130408 - 0.6209949923i\)
\(L(\frac12)\) \(\approx\) \(0.4608130408 - 0.6209949923i\)
\(L(1)\) \(\approx\) \(0.5126649204 - 0.3627181627i\)
\(L(1)\) \(\approx\) \(0.5126649204 - 0.3627181627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - iT \)
17 \( 1 + iT \)
19 \( 1 \)
23 \( 1 + iT \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 \)
41 \( 1 + T \)
43 \( 1 \)
47 \( 1 \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - iT \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 - iT \)
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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

−22.07104315002090413471643645576, −21.421207801691470620297572635982, −19.94735702930759696681614071222, −18.93796384021992827577049772721, −18.30722114500940334927058744195, −17.63999253406848531101081372018, −17.15064846411208408626388567668, −16.02013314222953958817759436660, −15.57478717536382627077042278787, −14.797212154939235214621652301426, −13.807485104977663856619118813407, −13.15544336286772719013606809860, −12.11214070535146954207454388611, −11.19563272508423226485374684101, −10.44053488773943063568832655653, −9.639052490249624484521264446251, −8.61406592604373879429728725832, −7.42050466272385359509182709334, −6.88631467503051616750986705645, −6.2015498586975722165973864703, −5.36892965162943216590627963678, −4.43807413807585319820251024769, −3.55025274284193280714776956567, −2.04942211841199842963417059221, −0.46476972629299769561378432035, 0.4204800416065974055660167811, 1.41301891087587305565588866824, 2.27579942002309684741870220354, 4.034887301403260310839638953251, 4.27574848895269687050944922654, 5.48695327609349080300305987994, 5.9565916929922609707815310610, 7.47583199561230242303149400793, 8.39566448181983589497103072120, 9.39041090655650824018588526994, 9.98951821219568718924289172741, 10.99272686843589347765051407451, 11.61675332344257478281430921227, 12.40328667356680007780517134218, 12.95100524209596681102204036189, 13.68703920321075891698333075110, 14.86231397345654279843391853961, 16.01909021261487138718118658153, 16.64287323428662124828493912121, 17.4792520382149684162253042044, 18.0401759341672142825259497453, 18.90696992482814844608066656505, 19.76593671954279410829708309946, 20.58475796010146504816363142652, 21.13243048663768171721456903720

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