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Properties

Label 1-2001-2001.893-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.914 + 0.403i$
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  + (−0.882 − 0.470i)2-s + (0.557 + 0.830i)4-s + (0.794 − 0.607i)5-s + (−0.986 − 0.162i)7-s + (−0.101 − 0.994i)8-s + (−0.986 + 0.162i)10-s + (−0.999 + 0.0407i)11-s + (−0.818 + 0.574i)13-s + (0.794 + 0.607i)14-s + (−0.377 + 0.925i)16-s + (−0.142 + 0.989i)17-s + (−0.557 − 0.830i)19-s + (0.947 + 0.320i)20-s + (0.900 + 0.433i)22-s + (0.262 − 0.965i)25-s + (0.992 − 0.122i)26-s + ⋯
L(s)  = 1  + (−0.882 − 0.470i)2-s + (0.557 + 0.830i)4-s + (0.794 − 0.607i)5-s + (−0.986 − 0.162i)7-s + (−0.101 − 0.994i)8-s + (−0.986 + 0.162i)10-s + (−0.999 + 0.0407i)11-s + (−0.818 + 0.574i)13-s + (0.794 + 0.607i)14-s + (−0.377 + 0.925i)16-s + (−0.142 + 0.989i)17-s + (−0.557 − 0.830i)19-s + (0.947 + 0.320i)20-s + (0.900 + 0.433i)22-s + (0.262 − 0.965i)25-s + (0.992 − 0.122i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5713940834 + 0.1204534587i\)
\(L(\frac12)\) \(\approx\) \(0.5713940834 + 0.1204534587i\)
\(L(1)\) \(\approx\) \(0.5927892163 - 0.1262197830i\)
\(L(1)\) \(\approx\) \(0.5927892163 - 0.1262197830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.882 - 0.470i)T \)
5 \( 1 + (0.794 - 0.607i)T \)
7 \( 1 + (-0.986 - 0.162i)T \)
11 \( 1 + (-0.999 + 0.0407i)T \)
13 \( 1 + (-0.818 + 0.574i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.557 - 0.830i)T \)
31 \( 1 + (0.339 - 0.940i)T \)
37 \( 1 + (-0.970 + 0.242i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.339 - 0.940i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.488 + 0.872i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (0.523 + 0.852i)T \)
67 \( 1 + (0.999 + 0.0407i)T \)
71 \( 1 + (-0.182 - 0.983i)T \)
73 \( 1 + (0.996 + 0.0815i)T \)
79 \( 1 + (0.377 + 0.925i)T \)
83 \( 1 + (0.0203 + 0.999i)T \)
89 \( 1 + (-0.714 - 0.699i)T \)
97 \( 1 + (0.768 + 0.639i)T \)
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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

−19.654635379879214750844280696852, −19.02043474775416769883151154215, −18.32668845824543529606289826688, −17.858856216622165590391704885084, −17.04572864804299118194663356499, −16.28880757535097559659868183599, −15.69138424012913364622600364679, −14.899147887149677700810388473566, −14.24008171545855737124382681508, −13.34900875115144991758343867893, −12.6262016822835071299286957628, −11.58803341925691258261415268793, −10.61441083773024837008665433472, −9.98030365392059465603416425196, −9.74266539948110335492498965502, −8.668832102202279881554336520647, −7.87312514318428582952879775213, −6.92555411911410838724101931544, −6.5363446823459956646910111899, −5.50324843558177459637311273975, −5.065805842262845785545881735776, −3.26928514186853679674805547666, −2.6299519320493987743174284129, −1.84535153743123165681178341573, −0.34898769816408147494883683050, 0.760452414496294577437790988910, 2.0994887005171002394616395950, 2.455416712611699407065272956841, 3.62097849017294941298543982282, 4.575460907836046437646629665455, 5.63248559343194740734746063768, 6.55590218231611721382813710421, 7.17775749852035434281828000175, 8.25002825854348808302089374429, 8.89921976863636172076298154234, 9.64316811898340576109332273835, 10.20751469204636380389032924873, 10.79827348635901267991957078842, 11.95318671874378527885210877772, 12.6536181803820673093251400882, 13.139853993765536626538702722093, 13.834406901051942437928928837, 15.21208451775270308506916401075, 15.76788928643745445172433629771, 16.744910082247860314678968627016, 17.047015538763533761597332648559, 17.74506167671882182094682137886, 18.65528325266454357294110621524, 19.30064858255408739776475423251, 19.84488908066007609377724397876

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