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Properties

Label 1-671-671.147-r0-0-0
Degree $1$
Conductor $671$
Sign $0.634 - 0.772i$
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 + 3-s + (0.669 − 0.743i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.5 + 0.866i)7-s + (0.309 − 0.951i)8-s + 9-s + (0.669 − 0.743i)10-s + (0.669 − 0.743i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (0.913 − 0.406i)15-s + (−0.104 − 0.994i)16-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)18-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + 3-s + (0.669 − 0.743i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.5 + 0.866i)7-s + (0.309 − 0.951i)8-s + 9-s + (0.669 − 0.743i)10-s + (0.669 − 0.743i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)14-s + (0.913 − 0.406i)15-s + (−0.104 − 0.994i)16-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.634 - 0.772i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.634 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.474758258 - 1.642108176i\)
\(L(\frac12)\) \(\approx\) \(3.474758258 - 1.642108176i\)
\(L(1)\) \(\approx\) \(2.470427426 - 0.7749996516i\)
\(L(1)\) \(\approx\) \(2.470427426 - 0.7749996516i\)

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 \( 1 + T \)
5 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.978 - 0.207i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−22.87401729935546747734826219144, −21.977684120321729917631777918798, −21.36565108979745929518130335331, −20.55932656206895466459210523524, −19.874474476234351721144503286440, −18.97547374627276127838525770477, −17.83071767224981306394400669460, −16.92168296091426395869830459530, −16.19779701957200065819202277869, −15.13890864530165861607115418681, −14.400112204602024010158776930983, −13.85493849290681544389461576628, −13.20536331850266651417319065224, −12.43978769436091867174992175362, −11.14751868232837617622072923416, −10.02638660147150981866581321003, −9.47684100264902966283936119326, −8.144104173851519593629416584536, −7.164502617638906389997745466409, −6.75734870798325142586899465240, −5.507539807386675400530983802958, −4.45297877677487804110293066198, −3.48589024484308890518851073222, −2.673312185313314417748417666002, −1.75864792245047277594099069427, 1.4353493959307794865263774698, 2.4966006865671203376047382247, 2.86190051709591106545975488274, 4.295477622517167574963990851267, 5.07809086570210353480389479062, 6.16466441139136038303132971814, 6.90917313368870840591844561101, 8.344664177178490968789090638997, 9.23686577702300850859117342355, 9.91470664138307179763349291175, 10.7727559405480923945820586178, 12.238780237141509818616280871088, 12.85394846884400068351226188625, 13.25491468499594158874535849574, 14.52164654128770011558780782368, 14.77977716854950210660296276323, 15.77002602837056209683692244991, 16.66875752846705140804834741900, 17.9298982812614812031495638581, 18.881711487240887172388886128587, 19.67164753347334651693448890952, 20.19901838187542178350054814432, 21.2460142860967975597862727076, 21.77424868142067291493615601807, 22.17380911704967508551244448809

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