↑

Properties

Label 1-1001-1001.194-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.751 + 0.659i$
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 + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.309 + 0.951i)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 + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.104 − 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)20-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.309 + 0.951i)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 + (0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.104 − 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07997307365 - 0.2125045006i\)
\(L(\frac12)\) \(\approx\) \(-0.07997307365 - 0.2125045006i\)
\(L(1)\) \(\approx\) \(0.6801025146 - 0.3158538266i\)
\(L(1)\) \(\approx\) \(0.6801025146 - 0.3158538266i\)

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 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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

−22.42310285655633378674827205368, −21.63076970616632810905479592260, −20.74994525611837979011003499502, −19.81518768262104229468142571805, −18.77174135709833541627521991586, −18.12935144362035234895684430947, −17.235307986516258398589053642327, −16.3394696073478338934736615424, −16.10670435860512789208327912311, −15.16772992164325252291210757371, −14.21126192568970924837006715790, −13.41149671091699408538805598989, −12.3486262356732962692852695205, −12.09426670749836553826911976246, −11.311453922458245888254978148541, −10.25095095220017615647841649396, −8.93133525808398130499037113641, −7.87400340145482206083782519710, −7.42150222017451910290039740616, −6.57495065350702186528902846604, −5.5535000647526669059505161934, −4.97775099492757383247201544241, −3.98409824853628315627935059387, −3.14205204892777059691960700889, −1.51451634888068864830586147280, 0.09462079218329048534022657985, 1.30485312030714227535857713848, 2.77337549402075497864178512688, 3.74473328074075156950947191686, 4.40577668129908669767918116712, 5.198829473738535195108116622403, 6.20230027309809150608732146755, 6.93209135247560491710081379636, 8.1786976296212733423472141547, 9.36139232714986437106768801417, 10.23850544947037101409235996286, 10.9793761291737776983044060925, 11.47954845403517311792334612302, 12.40063922225604028724414505813, 12.76232664448683055756943807892, 14.109612033169101631607275639046, 14.87653365530905741507775748129, 15.61893676049288945217233945882, 16.20922606509003429178806148129, 17.28963097953501070220442404999, 18.23305298596389877284176239096, 18.929501643219726149815935250548, 19.72940133656722163691730721215, 20.46393930128202591370869542199, 21.38618628383407791198458822810

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