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Properties

Label 1-1001-1001.2-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.120 + 0.992i$
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  + (0.587 − 0.809i)2-s + (−0.978 + 0.207i)3-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (−0.406 + 0.913i)6-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.207 + 0.978i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.207 − 0.978i)18-s + (−0.743 + 0.669i)19-s + (−0.994 − 0.104i)20-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.978 + 0.207i)3-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (−0.406 + 0.913i)6-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.207 + 0.978i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.207 − 0.978i)18-s + (−0.743 + 0.669i)19-s + (−0.994 − 0.104i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.120 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1196272746 - 0.1349617498i\)
\(L(\frac12)\) \(\approx\) \(-0.1196272746 - 0.1349617498i\)
\(L(1)\) \(\approx\) \(0.6461473177 - 0.4715925585i\)
\(L(1)\) \(\approx\) \(0.6461473177 - 0.4715925585i\)

Euler product

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

−22.37555374282508061143169904856, −21.7113662349662064894910776058, −21.212791539641625940709392885020, −19.838024193209890782256154034249, −18.7535308443092235518948512012, −17.907905098294235236640361907471, −17.63583682910617382262804946238, −16.7463561424562809493264742087, −15.78768212876688686933768626935, −15.355375260862945522901556377451, −14.184510212799401241743957611303, −13.66811108617548587261082721521, −12.76660253141013526222196409926, −11.95831553860945316971227709275, −11.13410804933533678840715818531, −10.363366436725335400848232253511, −9.28819979598851335404761563512, −8.15578666126437955629488786101, −7.06980784090329598546308892914, −6.664965759219060835824278063997, −5.92272741810410020262729543733, −5.011471271759309699584522920128, −4.206796478167254659694993636327, −3.001928383630127183415385931275, −1.94418577741353314116509708577, 0.06865576488225971440762524504, 1.40112537295344697790361934199, 2.13815118594126318307635958384, 3.771246351965553957805595912434, 4.40281351867715092364983542658, 5.22992512213916094118128006553, 5.97458567853148346160323533349, 6.669646458407849530777396065925, 8.31532964712715930051994122033, 9.168922477497508374197799948783, 10.22610811636847939916391793567, 10.49180400702080923690338463669, 11.818003431229186900353003373042, 12.08072754051265465320554174470, 13.07459089506730525839814787556, 13.52431846864214872503418896524, 14.746018676980116217512469208687, 15.55390508650575833178893175272, 16.41588641370259897267687581326, 17.21580350312959535257856167209, 17.955826646834948915062491313636, 18.734524472629721643645996654936, 19.78393815773262726112672369822, 20.42842414040279367293022555325, 21.32243333668497876392541473385

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