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Properties

Label 1-2001-2001.674-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.813 + 0.581i$
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  + (−0.557 − 0.830i)2-s + (−0.377 + 0.925i)4-s + (0.262 − 0.965i)5-s + (−0.947 − 0.320i)7-s + (0.979 − 0.202i)8-s + (−0.947 + 0.320i)10-s + (0.996 − 0.0815i)11-s + (0.339 − 0.940i)13-s + (0.262 + 0.965i)14-s + (−0.714 − 0.699i)16-s + (−0.959 − 0.281i)17-s + (0.377 − 0.925i)19-s + (0.794 + 0.607i)20-s + (−0.623 − 0.781i)22-s + (−0.862 − 0.505i)25-s + (−0.970 + 0.242i)26-s + ⋯
L(s)  = 1  + (−0.557 − 0.830i)2-s + (−0.377 + 0.925i)4-s + (0.262 − 0.965i)5-s + (−0.947 − 0.320i)7-s + (0.979 − 0.202i)8-s + (−0.947 + 0.320i)10-s + (0.996 − 0.0815i)11-s + (0.339 − 0.940i)13-s + (0.262 + 0.965i)14-s + (−0.714 − 0.699i)16-s + (−0.959 − 0.281i)17-s + (0.377 − 0.925i)19-s + (0.794 + 0.607i)20-s + (−0.623 − 0.781i)22-s + (−0.862 − 0.505i)25-s + (−0.970 + 0.242i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2178057562 - 0.6793869473i\)
\(L(\frac12)\) \(\approx\) \(-0.2178057562 - 0.6793869473i\)
\(L(1)\) \(\approx\) \(0.5211556201 - 0.4847037242i\)
\(L(1)\) \(\approx\) \(0.5211556201 - 0.4847037242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.557 - 0.830i)T \)
5 \( 1 + (0.262 - 0.965i)T \)
7 \( 1 + (-0.947 - 0.320i)T \)
11 \( 1 + (0.996 - 0.0815i)T \)
13 \( 1 + (0.339 - 0.940i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.377 - 0.925i)T \)
31 \( 1 + (-0.768 - 0.639i)T \)
37 \( 1 + (-0.882 + 0.470i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (0.768 - 0.639i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.523 + 0.852i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.452 - 0.891i)T \)
67 \( 1 + (-0.996 - 0.0815i)T \)
71 \( 1 + (0.933 - 0.359i)T \)
73 \( 1 + (0.986 + 0.162i)T \)
79 \( 1 + (0.714 - 0.699i)T \)
83 \( 1 + (-0.999 + 0.0407i)T \)
89 \( 1 + (0.0203 + 0.999i)T \)
97 \( 1 + (-0.182 - 0.983i)T \)
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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

−19.95067781809283570817795282829, −19.4510832068394981769123549356, −18.8373976226867121004727152320, −18.17853592237637512876320816655, −17.54693816968216207532098620976, −16.62536120270668112017819459928, −16.139764767813116902850463589086, −15.32710750613284410302390170995, −14.58539677531581990200714925624, −14.05417941129377127261597360342, −13.32854986006382149450833458079, −12.268141871416064464788913888018, −11.29325460678015414710416719072, −10.6088915874443272347055666357, −9.708608195533500346907461313045, −9.24715637687383943375793966351, −8.5128294290038171023792308362, −7.33594554938276339268814179464, −6.720207581555308131685397095048, −6.26916703925524097847193949974, −5.54144311932927458865022194270, −4.21210140978377336530712671845, −3.5217272868845445768959819214, −2.24895929608334613243447545481, −1.42275632370072279082273630982, 0.33502327668190332794483324894, 1.10098210717203184503020898485, 2.16475712289697246459804564528, 3.15038224906284530977179241948, 3.94100173602565138504628907196, 4.69840601139844581438653199978, 5.77744277963929249101239041631, 6.74981784084674736664933915052, 7.59695095399053053177076801759, 8.57963175053121416040035445331, 9.297082085432837208106116613370, 9.52445868517129291178396548672, 10.66681795618096343496679242941, 11.222207735720496775574892123122, 12.32785635868992982119206413150, 12.647073417437906443624646016515, 13.54045939486007023383130913006, 13.88759686161898762247300999222, 15.493288531276496422511252307440, 15.98298782912474860460042692573, 16.88163377163829871647412305451, 17.35196774548263248867017735870, 17.972873578421942199153379939171, 19.02106962101102787979877921382, 19.62939837403121567093201740663

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