↑

Properties

Label 1-1001-1001.212-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.629 - 0.777i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.809 + 0.587i)3-s + (−0.104 + 0.994i)4-s + (−0.669 + 0.743i)5-s + (0.978 + 0.207i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.913 + 0.406i)18-s + (0.809 − 0.587i)19-s + (−0.669 − 0.743i)20-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.809 + 0.587i)3-s + (−0.104 + 0.994i)4-s + (−0.669 + 0.743i)5-s + (0.978 + 0.207i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (−0.913 + 0.406i)18-s + (0.809 − 0.587i)19-s + (−0.669 − 0.743i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08443492590 - 0.1770066807i\)
\(L(\frac12)\) \(\approx\) \(0.08443492590 - 0.1770066807i\)
\(L(1)\) \(\approx\) \(0.4558295984 + 0.02875746599i\)
\(L(1)\) \(\approx\) \(0.4558295984 + 0.02875746599i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.30117025337558206242206376660, −21.03357798220314381889112594200, −20.08089933936320814913051320192, −19.347246593174785294507989217022, −18.69762571664739941419569730809, −17.95954650275965655560645179563, −17.110086377538138716319902969285, −16.50205205071168875480679225278, −16.01353096279979921561643052502, −15.06051249365468792451456736629, −14.09000764928732040517001764305, −13.16166579942074289100988599353, −12.241831546405531595880288019178, −11.63168347470849109571699805705, −10.58355613709153866374100472086, −9.89187075831533460753467988651, −8.67000245598988086429091268456, −8.029486326206386254131311975956, −7.35116602145235551545440677519, −6.42481769454613382322709031559, −5.53959215427207437533584497748, −4.91879033467727049514575029303, −3.776440210949074014728292416260, −1.893981805068967104272646136698, −1.052594576192804523149650958089, 0.14487905397930059063418941034, 1.487319386389482734802572177533, 3.03064735085449891867350987550, 3.53115005281895233187651698534, 4.56698214688848972143608992496, 5.56641686168249702520599826753, 6.93241825188338396530311998917, 7.39663436649350814265323932878, 8.55024732096779021663646965864, 9.592024392690163651766747157517, 10.13501693653138603668176154213, 11.0185437792399337146552148967, 11.645850183014622375505293937002, 12.06313452231958733385098834427, 13.209662351439115973574180109322, 14.2909517766490443453947607289, 15.35207479054673438169896543467, 16.0504113479840692968533196691, 16.677202923393914314676486332686, 17.67977890495584663588863198194, 18.25876566022194858991341427575, 18.87323966781558701612275955184, 19.920246674182940830685706766944, 20.460163529684569869003648868, 21.48609219705408301648119577295

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