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Properties

Label 1-2001-2001.131-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.840 - 0.541i$
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  + (0.359 + 0.933i)2-s + (−0.742 + 0.670i)4-s + (0.882 + 0.470i)5-s + (−0.992 + 0.122i)7-s + (−0.891 − 0.452i)8-s + (−0.122 + 0.992i)10-s + (0.728 − 0.685i)11-s + (−0.947 + 0.320i)13-s + (−0.470 − 0.882i)14-s + (0.101 − 0.994i)16-s + (−0.281 + 0.959i)17-s + (0.670 + 0.742i)19-s + (−0.970 + 0.242i)20-s + (0.900 + 0.433i)22-s + (0.557 + 0.830i)25-s + (−0.639 − 0.768i)26-s + ⋯
L(s)  = 1  + (0.359 + 0.933i)2-s + (−0.742 + 0.670i)4-s + (0.882 + 0.470i)5-s + (−0.992 + 0.122i)7-s + (−0.891 − 0.452i)8-s + (−0.122 + 0.992i)10-s + (0.728 − 0.685i)11-s + (−0.947 + 0.320i)13-s + (−0.470 − 0.882i)14-s + (0.101 − 0.994i)16-s + (−0.281 + 0.959i)17-s + (0.670 + 0.742i)19-s + (−0.970 + 0.242i)20-s + (0.900 + 0.433i)22-s + (0.557 + 0.830i)25-s + (−0.639 − 0.768i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.840 - 0.541i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.840 - 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3048481069 + 1.036431403i\)
\(L(\frac12)\) \(\approx\) \(-0.3048481069 + 1.036431403i\)
\(L(1)\) \(\approx\) \(0.7608511115 + 0.7345432653i\)
\(L(1)\) \(\approx\) \(0.7608511115 + 0.7345432653i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.359 + 0.933i)T \)
5 \( 1 + (0.882 + 0.470i)T \)
7 \( 1 + (-0.992 + 0.122i)T \)
11 \( 1 + (0.728 - 0.685i)T \)
13 \( 1 + (-0.947 + 0.320i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.670 + 0.742i)T \)
31 \( 1 + (0.607 + 0.794i)T \)
37 \( 1 + (-0.983 - 0.182i)T \)
41 \( 1 + (-0.755 - 0.654i)T \)
43 \( 1 + (-0.607 + 0.794i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.714 + 0.699i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.999 + 0.0203i)T \)
67 \( 1 + (-0.685 + 0.728i)T \)
71 \( 1 + (-0.862 - 0.505i)T \)
73 \( 1 + (-0.998 + 0.0611i)T \)
79 \( 1 + (-0.994 + 0.101i)T \)
83 \( 1 + (-0.917 - 0.396i)T \)
89 \( 1 + (-0.202 + 0.979i)T \)
97 \( 1 + (-0.965 - 0.262i)T \)
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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

−19.94375495010054756926973976418, −18.920672562403207673850545554065, −18.15240050994441210406034051196, −17.3760121113875860070352627643, −16.882355674595795273508856923386, −15.76761463373904891691585498003, −15.000057629911061424152134498469, −14.09573383285288583073544571902, −13.45342680443002613384034858349, −12.98639956286881822377365639230, −12.03913257420137796574257842587, −11.72361987092466256043363138418, −10.310624089619141056429519126193, −9.93860598906780077839300282514, −9.31308238940035734543748502580, −8.733917257392684573373387507933, −7.21820727752529327915595261230, −6.55026705915115266635289115014, −5.52509932761797577400888946907, −4.91524167361666286123070296122, −4.09827968895958101666067034876, −2.97904998143414818207443903767, −2.41314885851115584191949717525, −1.397838968300283147525371567680, −0.32184763354943110439572124827, 1.45032612076739959837667205954, 2.76818243642936624860518963695, 3.410855579146946949461121058909, 4.31185949621770945803613724155, 5.484396631409661947377244745681, 5.97627764392252217197155344544, 6.721828565319148719545962856566, 7.22321230294174215587768220203, 8.44566959298573057394851191704, 9.10744742351423832656994873157, 9.83377445871578436998459962110, 10.477407792920869512567721870758, 11.82249677721766667381327412279, 12.43940609776448140725645134374, 13.25834721107523011826927728451, 13.95634163095306192392712047255, 14.42595261426347300093519495992, 15.23040152692786661931047241044, 16.04911184529951035732344061751, 16.7942048964566939697682901373, 17.240639754164923096588342833864, 18.00263145798813506476495132164, 18.9851008225095142208238369426, 19.29360794255270959199833711055, 20.50488468235471971784804856171

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