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Properties

Label 1-2001-2001.1079-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.686 - 0.726i$
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  + (0.794 − 0.607i)2-s + (0.262 − 0.965i)4-s + (0.339 − 0.940i)5-s + (0.301 − 0.953i)7-s + (−0.377 − 0.925i)8-s + (−0.301 − 0.953i)10-s + (0.452 + 0.891i)11-s + (0.685 + 0.728i)13-s + (−0.339 − 0.940i)14-s + (−0.862 − 0.505i)16-s + (0.654 + 0.755i)17-s + (0.262 − 0.965i)19-s + (−0.818 − 0.574i)20-s + (0.900 + 0.433i)22-s + (−0.768 − 0.639i)25-s + (0.986 + 0.162i)26-s + ⋯
L(s)  = 1  + (0.794 − 0.607i)2-s + (0.262 − 0.965i)4-s + (0.339 − 0.940i)5-s + (0.301 − 0.953i)7-s + (−0.377 − 0.925i)8-s + (−0.301 − 0.953i)10-s + (0.452 + 0.891i)11-s + (0.685 + 0.728i)13-s + (−0.339 − 0.940i)14-s + (−0.862 − 0.505i)16-s + (0.654 + 0.755i)17-s + (0.262 − 0.965i)19-s + (−0.818 − 0.574i)20-s + (0.900 + 0.433i)22-s + (−0.768 − 0.639i)25-s + (0.986 + 0.162i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.686 - 0.726i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.686 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246733206 - 2.894070089i\)
\(L(\frac12)\) \(\approx\) \(1.246733206 - 2.894070089i\)
\(L(1)\) \(\approx\) \(1.470684866 - 1.215675816i\)
\(L(1)\) \(\approx\) \(1.470684866 - 1.215675816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.794 - 0.607i)T \)
5 \( 1 + (0.339 - 0.940i)T \)
7 \( 1 + (0.301 - 0.953i)T \)
11 \( 1 + (0.452 + 0.891i)T \)
13 \( 1 + (0.685 + 0.728i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (0.262 - 0.965i)T \)
31 \( 1 + (0.0611 - 0.998i)T \)
37 \( 1 + (0.947 + 0.320i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.0611 - 0.998i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.933 + 0.359i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.742 + 0.670i)T \)
67 \( 1 + (0.452 - 0.891i)T \)
71 \( 1 + (-0.970 - 0.242i)T \)
73 \( 1 + (0.591 + 0.806i)T \)
79 \( 1 + (-0.862 + 0.505i)T \)
83 \( 1 + (-0.523 + 0.852i)T \)
89 \( 1 + (-0.488 - 0.872i)T \)
97 \( 1 + (-0.992 - 0.122i)T \)
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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

−20.587097687554767255681681353621, −19.36198579957366745557654703051, −18.57167927322023251238523258942, −18.02064173677177417519267091152, −17.35621184216864213403883008475, −16.17404605105860389457558949434, −15.94657822616496667103231023772, −14.89369663150208533439852255608, −14.38471986330539030035920146886, −13.90967737720313983478589552700, −12.96748665309879358089878829464, −12.187018756369275576830142863901, −11.41620568356688174858647963649, −10.8803328551402675562841470943, −9.738619871631430995796602196332, −8.80875306658150925307908746301, −8.0419958001157017312101033547, −7.35422276094708592698996028632, −6.256420200581722874900100805182, −5.87146911871609985650258489319, −5.247534358840249587939384353864, −3.99880220609212619011532667080, −3.11288231686904050818946059721, −2.71757809544832831219499212339, −1.42262840325476646669118137440, 0.87917641594569765516586508711, 1.51136619463062063029761130587, 2.36089279427603245692985686772, 3.74547631258742574642840312274, 4.250474407567852395002646248089, 4.88352385308337461984956511716, 5.82775939331085304354365336093, 6.62394591975145347424330090825, 7.48930358776582308079237473196, 8.56877410859385705770645346187, 9.53399992366298857360600514680, 9.94866090538562392027967537036, 10.99199139307499300763596660398, 11.575964865959303276627048010937, 12.43371253018551864175284742160, 13.07948466322613188194699655943, 13.69112645256397781287498180586, 14.33094383522556029820450789691, 15.134697371281709490267061244628, 15.98547771826659513926593488177, 16.83296081796428658371714902934, 17.362273369991411119723959148018, 18.3162429951528065144283908222, 19.23045331999806523496138276612, 20.058649608514581732348505942115

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