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Properties

Label 1-1001-1001.256-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.463 - 0.885i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (0.309 − 0.951i)2-s + (−0.104 − 0.994i)3-s + (−0.809 − 0.587i)4-s + (−0.978 − 0.207i)5-s + (−0.978 − 0.207i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.104 + 0.994i)18-s + (0.913 + 0.406i)19-s + (0.669 + 0.743i)20-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.104 − 0.994i)3-s + (−0.809 − 0.587i)4-s + (−0.978 − 0.207i)5-s + (−0.978 − 0.207i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.104 + 0.994i)18-s + (0.913 + 0.406i)19-s + (0.669 + 0.743i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.463 - 0.885i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.463 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7816499182 - 0.4730172146i\)
\(L(\frac12)\) \(\approx\) \(0.7816499182 - 0.4730172146i\)
\(L(1)\) \(\approx\) \(0.6540287334 - 0.5503440396i\)
\(L(1)\) \(\approx\) \(0.6540287334 - 0.5503440396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−22.254660894944870022766992962421, −21.01762805312904941667619921455, −20.513366250016794499939712753883, −19.398916870245093936092756098379, −18.55884273526557486825209788488, −17.625009752228191895152979187788, −16.79715465629640391628350514791, −16.062544287153792649759367372060, −15.59809416652667973915255011392, −14.878327490722194153986470511982, −14.19299104404349307955207460354, −13.287126840052143700377153736405, −12.07714988521341780706372249479, −11.5281933633034382067528838772, −10.53135330947594200818360052195, −9.39698312255782396085017732860, −8.88842692402710342480276475096, −7.770486723709286071851957083232, −7.20206622086643725587688602150, −6.07031491786834011591252833896, −5.08136381874161862639843861858, −4.53382367991303192651804857598, −3.50463572348742181746913111491, −2.95089894330298682171861305666, −0.46598144004992104663331671001, 0.96836533205929831321601193830, 1.75357512435949088395302342102, 3.07310764278296197092473926696, 3.654249787026646904033535765278, 4.911585120105455211691352760595, 5.64933495107724491254361594413, 6.817466002863286026277196486860, 7.75547457000545909490637287022, 8.53111497909103218766145679298, 9.35170914213487917383975128993, 10.690366135837134139170318618591, 11.22615170109847811587104588689, 12.08364971327920386233412695575, 12.6378630111743010850285446986, 13.21770903882922882561982526224, 14.363808331192147680703866968954, 14.79879174743612507471219394156, 16.0437969880455482510601160860, 16.9927835409984648990621475757, 17.91467765621715233751327485729, 18.66603733333372971251377079423, 19.23465269919808478425638123684, 19.92173022841605891872424739729, 20.46639203329444267905827623936, 21.50012808858700622577148080332

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