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Properties

Label 1-1001-1001.226-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.246 + 0.969i$
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  + (−0.406 + 0.913i)2-s + (−0.978 − 0.207i)3-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.587 − 0.809i)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 + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.743 + 0.669i)18-s + (0.743 + 0.669i)19-s + (−0.587 − 0.809i)20-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.978 − 0.207i)3-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.587 − 0.809i)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 + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.743 + 0.669i)18-s + (0.743 + 0.669i)19-s + (−0.587 − 0.809i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.246 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8122135143 + 0.6315610799i\)
\(L(\frac12)\) \(\approx\) \(0.8122135143 + 0.6315610799i\)
\(L(1)\) \(\approx\) \(0.7354065059 + 0.3113759062i\)
\(L(1)\) \(\approx\) \(0.7354065059 + 0.3113759062i\)

Euler product

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

−21.47589318795486082722736477402, −20.81458715695763027878529078577, −20.15112171907503263588509317057, −18.78793551824859148815055944275, −18.49543665128047474928535874958, −17.522666407693401983414345889857, −17.01703277916423112947600427071, −16.45399206836949692232143619922, −15.2648665580304460785298958985, −14.09385451421311190943929208778, −13.28814177755947801063159251282, −12.54611212749103689358683451697, −11.860492341829246487570978157852, −10.91808350003382095257101516059, −10.322374942631252413225479759615, −9.600355135741627412470434709417, −8.91642234124129790936221699378, −7.67107341774224522864892657315, −6.67153292177320883098293892243, −5.596799798908692633597955198350, −4.94692392732341171045312657026, −3.90549234121897305629861092365, −2.77597799397997029731385094125, −1.66564727545378899711666116080, −0.75788180619245948813033640243, 1.02204458766449320454363756139, 1.8132874110370234349424358051, 3.53659496477783650366163710872, 4.98176657452769892917140149369, 5.487997652130745116081003485156, 6.10629308576878402844944898577, 7.13649292498360337597244556772, 7.607591845934378577573869652459, 9.00460041255390148098125747056, 9.69681371929838582970677153423, 10.410995818215931225595853805284, 11.209961999601745541865765055530, 12.41144817114369031734744819187, 13.16627351581700337138688808926, 14.04281965726151327983852746791, 14.66687066458228759463411706004, 15.880386326688087985600902608930, 16.42005312533611027054789908635, 17.17023719061754841244530161977, 17.748680059670859388801426404786, 18.4836182062474879244925840808, 18.94418849732996239626859446218, 20.224346446482592406793542633103, 21.29141737614544555290163549566, 22.039172526021655478865211397598

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