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Properties

Label 1-671-671.16-r0-0-0
Degree $1$
Conductor $671$
Sign $0.995 + 0.0922i$
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  + (−0.978 + 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.913 + 0.406i)5-s + (0.913 + 0.406i)6-s + (0.913 + 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.978 − 0.207i)10-s + (−0.978 − 0.207i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.5 − 0.866i)15-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.913 + 0.406i)5-s + (0.913 + 0.406i)6-s + (0.913 + 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.978 − 0.207i)10-s + (−0.978 − 0.207i)12-s + (0.913 + 0.406i)13-s + (−0.978 − 0.207i)14-s + (−0.5 − 0.866i)15-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.995 + 0.0922i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.995 + 0.0922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015158004 + 0.04693436449i\)
\(L(\frac12)\) \(\approx\) \(1.015158004 + 0.04693436449i\)
\(L(1)\) \(\approx\) \(0.7912896041 + 0.01461777106i\)
\(L(1)\) \(\approx\) \(0.7912896041 + 0.01461777106i\)

Euler product

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

−22.630763613518504977884320947542, −21.566024457064435198771272755381, −21.022004931902064597439687397921, −20.58921784341756751247939537966, −19.52264255141154381207190219150, −18.15578578927972304839715866850, −17.83698807877220717653164657893, −17.15845220975515668407550668726, −16.461052757965919127957814967388, −15.6330259086783361057118034865, −14.69921132377093433740130945151, −13.46637920517774985881294107192, −12.50625908413410697247834591009, −11.47832519580365803406436507569, −10.8741440495146995957672603452, −10.10427642122241693688000104625, −9.37923018515006803092721679054, −8.432932519608936761393355204285, −7.50114277455147089064316717286, −6.18199872693049284810846672906, −5.669267216307299733132993678, −4.43983375117008149168382061400, −3.31263277286798764807696705794, −1.695627123529766745233079074252, −1.03568125870252726797804075110, 1.07272040204281470583275114729, 1.85592470991517588125008519388, 2.8468837053987625431283398309, 5.01791544057782746111379187025, 5.57790053306448851580676169090, 6.69405701793262950042095655854, 7.10688579662386573858961872218, 8.36241142212580167055837852720, 9.10179099751231316906871821427, 10.23489580616359301845829187640, 11.00274280028918927366842171923, 11.55286136366287906859968057969, 12.53517381798117185132558976328, 13.85215611623751045436079412545, 14.36349318078033556684809030129, 15.71774637500654081940261303467, 16.34322753056118057390031925216, 17.39243078920619116576747052889, 17.89928852712020328417459219922, 18.41736349866031992681492902207, 19.01775628010061500824637998845, 20.37298432453375762520779881197, 21.04495567662963650550981450266, 21.94678965225011122207145604067, 22.837606356144887365326339330183

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