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Properties

Label 1-671-671.137-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.620 - 0.784i$
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  + (0.669 − 0.743i)2-s + 3-s + (−0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (0.669 − 0.743i)6-s + (−0.5 − 0.866i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.104 − 0.994i)10-s + (−0.104 − 0.994i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.669 − 0.743i)15-s + (−0.978 + 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.669 − 0.743i)18-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + 3-s + (−0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (0.669 − 0.743i)6-s + (−0.5 − 0.866i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.104 − 0.994i)10-s + (−0.104 − 0.994i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.669 − 0.743i)15-s + (−0.978 + 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.669 − 0.743i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.620 - 0.784i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.620 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293702733 - 2.672098061i\)
\(L(\frac12)\) \(\approx\) \(1.293702733 - 2.672098061i\)
\(L(1)\) \(\approx\) \(1.574577428 - 1.360070945i\)
\(L(1)\) \(\approx\) \(1.574577428 - 1.360070945i\)

Euler product

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

−23.01688274448746128616797083867, −22.07876850414996089678333785216, −21.64506967768314435842765991194, −20.91231472760363851013725683542, −19.83867375170465321051791984455, −18.89754983466680319541642967748, −18.03898779064564957394949306980, −17.45439713269319128269846991810, −15.94411830829600816209996414591, −15.48734457119098834894446667867, −14.89539510831898550811669189584, −13.81013469755050064939054485785, −13.43406565417359525236857540537, −12.63252735485856763469122782481, −11.45396148378590703510897967339, −10.25791269757807559454427477783, −9.18676105517990391900613321695, −8.58899307675306297245506839547, −7.60574500129503778189607509573, −6.45125654059339544735276333171, −6.12628113314842243784176491092, −4.76047278371085455342289843450, −3.63603196021131736596741858288, −2.800817201478777001898069437157, −2.1030160715655698726022711320, 1.06852871569474448663911862198, 1.952555364377265687714810184594, 2.97880076283950658667694839880, 4.13505109948422932316379279223, 4.506415639143161579001164207714, 6.05526735389301033166740139521, 6.72575335387829227968368218115, 8.259137742333982555057650549804, 9.008747384573374806345353300740, 9.954567468248358569876859859918, 10.42355350831763686469684867785, 11.71028982400171533627730773986, 12.91357485666324069937042365259, 13.205800486632653583446604786608, 13.97674840178076496536540396827, 14.59887582206111823893215713510, 15.87240290984919518739954601912, 16.40973190800440159416782084528, 17.810709910833428282452145264783, 18.62585731393697659940267754657, 19.61394914462262702510185762752, 20.16376891794998312494570247699, 20.79368611321991771397252721151, 21.38859359139018766728013667023, 22.30374660488592133268859448995

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