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Properties

Label 1-2001-2001.383-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.906 + 0.423i$
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  + (−0.0611 + 0.998i)2-s + (−0.992 − 0.122i)4-s + (0.996 − 0.0815i)5-s + (−0.0203 + 0.999i)7-s + (0.182 − 0.983i)8-s + (0.0203 + 0.999i)10-s + (0.377 − 0.925i)11-s + (−0.452 − 0.891i)13-s + (−0.996 − 0.0815i)14-s + (0.970 + 0.242i)16-s + (−0.841 + 0.540i)17-s + (−0.992 − 0.122i)19-s + (−0.999 − 0.0407i)20-s + (0.900 + 0.433i)22-s + (0.986 − 0.162i)25-s + (0.917 − 0.396i)26-s + ⋯
L(s)  = 1  + (−0.0611 + 0.998i)2-s + (−0.992 − 0.122i)4-s + (0.996 − 0.0815i)5-s + (−0.0203 + 0.999i)7-s + (0.182 − 0.983i)8-s + (0.0203 + 0.999i)10-s + (0.377 − 0.925i)11-s + (−0.452 − 0.891i)13-s + (−0.996 − 0.0815i)14-s + (0.970 + 0.242i)16-s + (−0.841 + 0.540i)17-s + (−0.992 − 0.122i)19-s + (−0.999 − 0.0407i)20-s + (0.900 + 0.433i)22-s + (0.986 − 0.162i)25-s + (0.917 − 0.396i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.906 + 0.423i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.906 + 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451806303 + 0.3223405566i\)
\(L(\frac12)\) \(\approx\) \(1.451806303 + 0.3223405566i\)
\(L(1)\) \(\approx\) \(0.9925557859 + 0.3916922021i\)
\(L(1)\) \(\approx\) \(0.9925557859 + 0.3916922021i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0611 + 0.998i)T \)
5 \( 1 + (0.996 - 0.0815i)T \)
7 \( 1 + (-0.0203 + 0.999i)T \)
11 \( 1 + (0.377 - 0.925i)T \)
13 \( 1 + (-0.452 - 0.891i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (-0.992 - 0.122i)T \)
31 \( 1 + (0.591 - 0.806i)T \)
37 \( 1 + (0.685 - 0.728i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.591 - 0.806i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.794 - 0.607i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.262 + 0.965i)T \)
67 \( 1 + (0.377 + 0.925i)T \)
71 \( 1 + (0.818 - 0.574i)T \)
73 \( 1 + (0.714 - 0.699i)T \)
79 \( 1 + (0.970 - 0.242i)T \)
83 \( 1 + (0.557 + 0.830i)T \)
89 \( 1 + (-0.882 - 0.470i)T \)
97 \( 1 + (-0.301 - 0.953i)T \)
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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

−20.032510967031922436720095462245, −19.3130891886523538373406918968, −18.472340245120872148197203230647, −17.72906185668838822334581574240, −17.119944342429626083656675610389, −16.76200004652694471041080385447, −15.38051250304050770695242423555, −14.249126272221652933094218203537, −14.098531783757077589195708322640, −13.18042950559560108903955032149, −12.60785350120014379675183487032, −11.72380941505431909933830029832, −10.871083604401568988013266369923, −10.279338257884352500602222455809, −9.54728833169075020293165616059, −9.09635777790933339504171833366, −8.011793184745267917364537464, −6.91018168371121078523800760070, −6.40173947915672451886255888989, −4.9377714621109732618951891895, −4.560932019990024607765436836490, −3.67057929992625984815474075004, −2.46545615008963596363346587772, −1.93201740910456876366473116666, −0.9968823931254339473362230079, 0.604134559045198233379149989484, 1.96697118849621454640841810644, 2.837694249399770632068460619216, 4.02634726929614187418385411057, 5.00457650933667955319294241143, 5.783825391131381560022410234195, 6.15274280354971951309351750807, 6.9615561359862626767190505919, 8.23219421646343466817604367053, 8.621367356817409007732608963056, 9.34670298094815643044078343751, 10.11965064826578598830145094002, 10.939325814914212242425808961280, 12.1003995659805144656796163204, 13.00771681628330895345129773001, 13.35826376783733089504960615160, 14.28634665787202389758504765357, 15.10112357455562208883314341322, 15.36903246622062715930650419891, 16.626794081202641950709649700493, 16.90286158802009337901388401502, 17.88349174279090585368740134990, 18.18897729354980481505276712424, 19.14424399455213654261011067934, 19.73658934815561780857297475701

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