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Properties

Label 1-2001-2001.689-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.936 - 0.349i$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.222 + 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.623 − 0.781i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.222 + 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.623 − 0.781i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.936 - 0.349i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.936 - 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01407746836 - 0.07799614601i\)
\(L(\frac12)\) \(\approx\) \(0.01407746836 - 0.07799614601i\)
\(L(1)\) \(\approx\) \(0.4875045439 + 0.04143084848i\)
\(L(1)\) \(\approx\) \(0.4875045439 + 0.04143084848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−20.02811832476182990033798378602, −19.28262075643976620808809431457, −19.11134665371199866633210465484, −18.285635979384901661071190099030, −17.418673165470693989654469662661, −16.45209342448047861462448551284, −16.12036271307875588157389564932, −15.582692895820556806354335189130, −14.5241048764974646792705525358, −13.47342123867032369594467407716, −12.58052796543825543969752152408, −12.02458895311414734615628618224, −11.406075362367213524233834475635, −10.77907375672184528931372894530, −9.54390143629280584562762662079, −9.14532224591726058272468108812, −8.40931791023384540087248888644, −7.76604388251676846694530386560, −6.72559161689730836667038496209, −6.13802952303048374214641525429, −4.85114988767079955756601585525, −3.82951648435316582684241611249, −3.1913542502884914735607794994, −2.22586536207445107887068918080, −1.15742286975357943816581899548, 0.04685043741440467075025825127, 1.031924280490848281657513491145, 2.402285980467786101574320677218, 3.191190667109929142875587602503, 4.30416075033903233602689821312, 5.08749567887288110084881377592, 6.42268817171804036305150041678, 6.88182071464967119513976453435, 7.55859733266610243055511469236, 8.18329720355860510292764728683, 9.20209050818478677975655258605, 9.961201524283419538617509877892, 10.53813835910289491092360830871, 11.30552891277112876267134830877, 12.0451182298424346572169068099, 13.02128426647917294090342022305, 13.89418742333147331359757401960, 14.85796185133947069870430786871, 15.51030406809748203936820774965, 15.78509062189958818714117439517, 16.866073254516201551039478296296, 17.44116814689443816915272363736, 18.10180436884212351859045473025, 18.887540873447891302249601235138, 19.78460724671043013906547386915

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