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Properties

Label 1-2001-2001.5-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.990 - 0.137i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (−0.714 + 0.699i)2-s + (0.0203 − 0.999i)4-s + (0.488 − 0.872i)5-s + (−0.262 + 0.965i)7-s + (0.685 + 0.728i)8-s + (0.262 + 0.965i)10-s + (−0.947 − 0.320i)11-s + (0.182 − 0.983i)13-s + (−0.488 − 0.872i)14-s + (−0.999 − 0.0407i)16-s + (−0.415 + 0.909i)17-s + (0.0203 − 0.999i)19-s + (−0.862 − 0.505i)20-s + (0.900 − 0.433i)22-s + (−0.523 − 0.852i)25-s + (0.557 + 0.830i)26-s + ⋯
L(s)  = 1  + (−0.714 + 0.699i)2-s + (0.0203 − 0.999i)4-s + (0.488 − 0.872i)5-s + (−0.262 + 0.965i)7-s + (0.685 + 0.728i)8-s + (0.262 + 0.965i)10-s + (−0.947 − 0.320i)11-s + (0.182 − 0.983i)13-s + (−0.488 − 0.872i)14-s + (−0.999 − 0.0407i)16-s + (−0.415 + 0.909i)17-s + (0.0203 − 0.999i)19-s + (−0.862 − 0.505i)20-s + (0.900 − 0.433i)22-s + (−0.523 − 0.852i)25-s + (0.557 + 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.990 - 0.137i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.990 - 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002418212176 + 0.03500612457i\)
\(L(\frac12)\) \(\approx\) \(0.002418212176 + 0.03500612457i\)
\(L(1)\) \(\approx\) \(0.6097973658 + 0.09134126711i\)
\(L(1)\) \(\approx\) \(0.6097973658 + 0.09134126711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.714 + 0.699i)T \)
5 \( 1 + (0.488 - 0.872i)T \)
7 \( 1 + (-0.262 + 0.965i)T \)
11 \( 1 + (-0.947 - 0.320i)T \)
13 \( 1 + (0.182 - 0.983i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.0203 - 0.999i)T \)
31 \( 1 + (0.933 + 0.359i)T \)
37 \( 1 + (-0.377 + 0.925i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.933 + 0.359i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.591 - 0.806i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (-0.301 - 0.953i)T \)
67 \( 1 + (-0.947 + 0.320i)T \)
71 \( 1 + (-0.101 - 0.994i)T \)
73 \( 1 + (-0.794 + 0.607i)T \)
79 \( 1 + (-0.999 + 0.0407i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (-0.996 - 0.0815i)T \)
97 \( 1 + (0.742 + 0.670i)T \)
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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

−19.43594040082079658430529552476, −18.76138375233202156659738344215, −18.27985478234984892331389356045, −17.556211769272680196585660036406, −16.814157882488013241656248766579, −16.17022891723205429619291121558, −15.32311633777827429322713619459, −14.131336250205010436945768294015, −13.65615973463351553843620378531, −13.00766845469933481643411224931, −11.93598132468316045042653862758, −11.29662295954158036502722599553, −10.40764635241094384991190539615, −10.122317531121678305080744299024, −9.35828931457688954607869789047, −8.38248163596965618047288189399, −7.365525448786619293612121872370, −7.05502402960564671092851211527, −6.09865302372729366010196525660, −4.78380456826461929531451432948, −3.90184850968655135205924057676, −3.08995739490405149458829798412, −2.26512861362525933385346805758, −1.46314400519014943354917709971, −0.01558350621432155423074533656, 1.19331723814830923705784528957, 2.19089778191277647576284907777, 3.06745974696084468080080196044, 4.706920797935814270437037199633, 5.23038248683532478892248526753, 5.95782400734510626958412393433, 6.574461872883781306040815045414, 7.81436812546619311595265294400, 8.51942899549346085476498682448, 8.8083721238580673689054573309, 9.86338729863033918211719081356, 10.36461164292296689986477607490, 11.31316888635461790866296093001, 12.330379358199940400090591830429, 13.20022685208270226460825169434, 13.55597856857494485148586628067, 14.8120191954077871409427895041, 15.56149674101732796044970195701, 15.79449490698996322995061440016, 16.73933078827407665186417076731, 17.547306870435619759393699561086, 17.91356347352202859426672756631, 18.77853714115316094948194860667, 19.451348383629932955569796818925, 20.22286707352579289381100214987

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