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Properties

Label 1-571-571.96-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.984 + 0.177i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.926 + 0.376i)2-s + (0.202 − 0.979i)3-s + (0.716 − 0.697i)4-s + (0.329 + 0.944i)5-s + (0.180 + 0.983i)6-s + (−0.0165 − 0.999i)7-s + (−0.401 + 0.915i)8-s + (−0.917 − 0.396i)9-s + (−0.660 − 0.750i)10-s + (−0.319 − 0.947i)11-s + (−0.537 − 0.843i)12-s + (−0.889 + 0.456i)13-s + (0.391 + 0.920i)14-s + (0.991 − 0.131i)15-s + (0.0275 − 0.999i)16-s + (0.565 + 0.824i)17-s + ⋯
L(s)  = 1  + (−0.926 + 0.376i)2-s + (0.202 − 0.979i)3-s + (0.716 − 0.697i)4-s + (0.329 + 0.944i)5-s + (0.180 + 0.983i)6-s + (−0.0165 − 0.999i)7-s + (−0.401 + 0.915i)8-s + (−0.917 − 0.396i)9-s + (−0.660 − 0.750i)10-s + (−0.319 − 0.947i)11-s + (−0.537 − 0.843i)12-s + (−0.889 + 0.456i)13-s + (0.391 + 0.920i)14-s + (0.991 − 0.131i)15-s + (0.0275 − 0.999i)16-s + (0.565 + 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.984 + 0.177i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.984 + 0.177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01342353321 - 0.1499276916i\)
\(L(\frac12)\) \(\approx\) \(0.01342353321 - 0.1499276916i\)
\(L(1)\) \(\approx\) \(0.5459271911 - 0.1151580537i\)
\(L(1)\) \(\approx\) \(0.5459271911 - 0.1151580537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.926 + 0.376i)T \)
3 \( 1 + (0.202 - 0.979i)T \)
5 \( 1 + (0.329 + 0.944i)T \)
7 \( 1 + (-0.0165 - 0.999i)T \)
11 \( 1 + (-0.319 - 0.947i)T \)
13 \( 1 + (-0.889 + 0.456i)T \)
17 \( 1 + (0.565 + 0.824i)T \)
19 \( 1 + (-0.868 - 0.495i)T \)
23 \( 1 + (-0.995 + 0.0990i)T \)
29 \( 1 + (-0.998 + 0.0550i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (0.159 - 0.987i)T \)
41 \( 1 + (-0.821 - 0.569i)T \)
43 \( 1 + (0.0935 + 0.995i)T \)
47 \( 1 + (0.851 + 0.523i)T \)
53 \( 1 + (0.970 - 0.240i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (-0.899 - 0.436i)T \)
67 \( 1 + (-0.693 - 0.720i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.775 + 0.631i)T \)
79 \( 1 + (-0.0385 + 0.999i)T \)
83 \( 1 + (0.761 - 0.648i)T \)
89 \( 1 + (-0.942 - 0.335i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−23.86443194475060251538237727243, −22.382836690606038229664914158178, −21.86227468338722759619653617144, −20.879085525387380979091549407347, −20.47846800419092543105507969638, −19.7534902021561585838772065472, −18.64766616608288142302230736574, −17.80774935846501179676254596459, −16.864573853251166526647221745626, −16.39613469604493714998042007985, −15.34066795050736393112611660446, −14.85445224608597556828880981939, −13.29301803982909220006038642791, −12.22885172063393324538444740253, −11.845479845156730813835214183573, −10.38763834460055551293367759755, −9.81242470634157718257422986864, −9.16193568786385508259157185915, −8.36035062599412289867254389027, −7.50841106662491707028586444490, −5.868157009311318701515297912, −5.070472307815866112400605734899, −3.96236106188982482473081031975, −2.605142294048233170389942862221, −1.932392177743499342320111212861, 0.09280985811954510498209855998, 1.60371999541731466910535832087, 2.468323847143969697904131517294, 3.65300219383489843173503557129, 5.60647897837658325719582700978, 6.36338589756757185262498977347, 7.21505494135980708335890328120, 7.72505847942263062609531998010, 8.75066351586018306028258962722, 9.84929760181974105151611425233, 10.7444197362306165019882750528, 11.32618509314640496934140165056, 12.60005646126061408589263394271, 13.75104001325555032774747155910, 14.350002176019636208202357011185, 15.03809317652953492061376500753, 16.46549039055777641650788913049, 17.10072827143938615819040564289, 17.84118856977240331595539865780, 18.63220177887118848849843462737, 19.36891366100059954925349132985, 19.775878918415350731355985069800, 20.985814232242109744265964803456, 21.9864468565467980228708536103, 23.24212489827537067197435363625

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