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Properties

Label 1-1001-1001.250-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.742 + 0.670i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (−0.309 + 0.951i)2-s + (−0.913 − 0.406i)3-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (0.913 − 0.406i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.913 + 0.406i)18-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.913 − 0.406i)3-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (0.913 − 0.406i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.913 + 0.406i)18-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.742 + 0.670i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.742 + 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5555887524 + 0.2137662642i\)
\(L(\frac12)\) \(\approx\) \(0.5555887524 + 0.2137662642i\)
\(L(1)\) \(\approx\) \(0.5286849440 + 0.2085022346i\)
\(L(1)\) \(\approx\) \(0.5286849440 + 0.2085022346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−21.29271999378986519132530827945, −20.84127513065192445630772500399, −20.19336319610415488180936205391, −19.15468814681610326285581549014, −18.547325659182774557542134926899, −17.68425224276364527217893779538, −16.7615657275319656196579380815, −16.45665472563536049711108727793, −15.460915980552187323069920381001, −14.41143626506909695189281748654, −13.17217795939716680510874894919, −12.566758646064755552983479089840, −11.7929397775993222177399731297, −11.31785379707038737776550653988, −10.40202090794295610808484070299, −9.57741482394477746099036907066, −8.86825688610910961256566474333, −7.86742529685869497485293445768, −6.95661200776151013688166107731, −5.44447437046460274799769096563, −4.91852840789141872242704712746, −3.9588434556471759092352117007, −3.24877720766555978409017932674, −1.65286326617634176965427506657, −0.69736558294668853090626256966, 0.55850509034340016433340928935, 1.924054009022507908445895394015, 3.54226700384786316719787834615, 4.52944891520087565025453187039, 5.4642086881681584405146439289, 6.27745900637390087134487058318, 7.10160694720393389484377383228, 7.54089372269334417870179744703, 8.55620040403338606715132918235, 9.601908444986344587007101247001, 10.82884184006618832972669308244, 10.94223211371613699609660985433, 12.26011037423607078833058231763, 13.03323632259500185844498881146, 13.91488527847108172045841466868, 15.02355988934307017384608991161, 15.34918550623356888798256639043, 16.43622854116153849059100659964, 16.95133001520504532258168604883, 17.85442004196161420381183446224, 18.39510060622732099310907457442, 19.2407299586280724686733384563, 19.62849947449573658133729706378, 21.24512646225275146308749887665, 22.19959974251930755049631250674

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