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Properties

Label 1-671-671.140-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.842 - 0.538i$
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  + (−0.994 − 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.207 − 0.978i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.994 + 0.104i)10-s + (−0.104 − 0.994i)12-s + (0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (−0.5 − 0.866i)15-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (0.866 − 0.5i)18-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.207 − 0.978i)7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.994 + 0.104i)10-s + (−0.104 − 0.994i)12-s + (0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (−0.5 − 0.866i)15-s + (0.913 + 0.406i)16-s + (0.743 − 0.669i)17-s + (0.866 − 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.842 - 0.538i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2301186861 - 0.7878082831i\)
\(L(\frac12)\) \(\approx\) \(0.2301186861 - 0.7878082831i\)
\(L(1)\) \(\approx\) \(0.5916776059 - 0.4111040891i\)
\(L(1)\) \(\approx\) \(0.5916776059 - 0.4111040891i\)

Euler product

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

−23.063218558612246859676110073470, −21.90805564765338811985741136983, −21.43061098528551819518254989716, −20.80324380356767027910959965522, −19.855475869955527256911994436556, −18.66185548336637544858773190442, −18.23835089103603905648069809357, −17.33200912022518800406476708192, −16.47839520642222724436084867840, −16.03644740175282025425917924794, −14.78561172595231300318365348361, −14.58557188220420773787976196027, −12.90795477461334359997842577785, −12.008410965530710685423135678454, −10.862797410202638911541984544942, −10.4819067962351111357298800448, −9.4439861926804709340457050197, −8.9585541894694813286026041278, −8.10112922336992123089870933263, −6.32632541351893112074256416355, −6.154684425642273897258761089222, −5.15816213859289493153578824430, −3.588447997412293314329788602917, −2.56266631748357575171409215526, −1.49364581057774316127386011120, 0.59004961892267822516518102333, 1.52827397707442019737554451230, 2.41405229805265160328866435144, 3.714823105623237542616898509700, 5.51079659002085884104249453075, 6.214906893917818268464123137277, 7.06014017927961957504959097982, 7.85394383024418635184514205652, 8.78142472274624986740839448252, 9.776683019908986278762452135313, 10.56893895113037721757068847104, 11.34162506770952763444326097409, 12.33974595685357488896645164556, 13.340326432762530202610775352649, 13.74626506897213577694947265051, 15.05256745080345634526249228176, 16.4452807143029582591565084888, 16.81253653145839697418808725220, 17.60418724411162139095733193624, 18.26691614438626373984501103909, 18.95053948715669414942298869439, 19.92705307817631272920742123800, 20.559045774775682035382392728948, 21.36486028215228137830355056648, 22.50895758286462102955048568386

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