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Properties

Label 1-671-671.126-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.581 + 0.813i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (−0.913 + 0.406i)2-s + (−0.809 + 0.587i)3-s + (0.669 − 0.743i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.913 + 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.104 − 0.994i)10-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.104 − 0.994i)15-s + (−0.104 − 0.994i)16-s + (0.5 + 0.866i)17-s + (0.104 + 0.994i)18-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (−0.809 + 0.587i)3-s + (0.669 − 0.743i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.913 + 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.104 − 0.994i)10-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.104 − 0.994i)15-s + (−0.104 − 0.994i)16-s + (0.5 + 0.866i)17-s + (0.104 + 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2237663044 + 0.4352540688i\)
\(L(\frac12)\) \(\approx\) \(0.2237663044 + 0.4352540688i\)
\(L(1)\) \(\approx\) \(0.4201255836 + 0.2507189057i\)
\(L(1)\) \(\approx\) \(0.4201255836 + 0.2507189057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−22.50638539263946087278008275195, −21.58735901250332259583046307444, −20.65058720643652851465403153498, −19.73750257391557389956578162964, −19.256361492608499420491967478459, −18.45992729333742736903505600062, −17.5599285017253837322280453379, −16.6147113690809819486574279529, −16.38890536571282444293755218570, −15.602246401661168200434644143511, −13.78774045729863346751727071449, −12.99340502972961741562858539259, −12.21986723141736814928440073858, −11.60791472459562297189446323710, −10.83795689580991248423915683519, −9.578189675980191275378154236373, −9.16304106587016336354599202301, −7.7397767338955351204782223565, −7.266798394199221762855384787057, −6.34337501142429332841986985108, −5.10279208234044688015198848325, −3.97244295164804977181683023458, −2.75426774238101291835179842885, −1.351246458137183662838368149894, −0.53403718522322955857220497515, 0.90431041848356384128187528135, 2.76519517834444368325917501422, 3.55677456022670600966977403346, 5.117092599128466982111375116506, 6.02560835304632400510363964601, 6.640116118747882792942786084223, 7.606457377598962928575961281581, 8.631819580147508103695625944785, 9.81099455264190037146007719723, 10.24471932130312972146714152567, 11.02269295587143451038557980663, 11.92554954144599613031238104567, 12.75194365372547893353985180737, 14.4429066944074602673457355313, 15.10324547038607884622310015714, 15.80446033631158201946619586595, 16.37655959655029376216157843727, 17.31529184006475734552928125473, 18.14371786417416645182568840265, 18.74530441478154411948385092684, 19.59786088352005032632633640788, 20.40861303311894353093090208449, 21.58583945309682517633372104322, 22.38088316408544011393306067668, 23.114553192810485672441064166591

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