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Properties

Label 1-571-571.176-r0-0-0
Degree $1$
Conductor $571$
Sign $0.441 - 0.897i$
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  + (−0.677 − 0.735i)2-s + (−0.277 + 0.960i)3-s + (−0.0825 + 0.996i)4-s + (−0.724 + 0.689i)5-s + (0.894 − 0.446i)6-s + (0.863 + 0.504i)7-s + (0.789 − 0.614i)8-s + (−0.846 − 0.533i)9-s + (0.997 + 0.0660i)10-s + (0.997 − 0.0660i)11-s + (−0.934 − 0.355i)12-s + (−0.0165 − 0.999i)13-s + (−0.213 − 0.976i)14-s + (−0.461 − 0.887i)15-s + (−0.986 − 0.164i)16-s + (−0.148 − 0.988i)17-s + ⋯
L(s)  = 1  + (−0.677 − 0.735i)2-s + (−0.277 + 0.960i)3-s + (−0.0825 + 0.996i)4-s + (−0.724 + 0.689i)5-s + (0.894 − 0.446i)6-s + (0.863 + 0.504i)7-s + (0.789 − 0.614i)8-s + (−0.846 − 0.533i)9-s + (0.997 + 0.0660i)10-s + (0.997 − 0.0660i)11-s + (−0.934 − 0.355i)12-s + (−0.0165 − 0.999i)13-s + (−0.213 − 0.976i)14-s + (−0.461 − 0.887i)15-s + (−0.986 − 0.164i)16-s + (−0.148 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5152002008 - 0.3206800251i\)
\(L(\frac12)\) \(\approx\) \(0.5152002008 - 0.3206800251i\)
\(L(1)\) \(\approx\) \(0.6283861028 - 0.04225173450i\)
\(L(1)\) \(\approx\) \(0.6283861028 - 0.04225173450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.677 - 0.735i)T \)
3 \( 1 + (-0.277 + 0.960i)T \)
5 \( 1 + (-0.724 + 0.689i)T \)
7 \( 1 + (0.863 + 0.504i)T \)
11 \( 1 + (0.997 - 0.0660i)T \)
13 \( 1 + (-0.0165 - 0.999i)T \)
17 \( 1 + (-0.148 - 0.988i)T \)
19 \( 1 + (-0.340 - 0.940i)T \)
23 \( 1 + (-0.999 - 0.0330i)T \)
29 \( 1 + (0.945 - 0.324i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (0.601 - 0.799i)T \)
41 \( 1 + (-0.879 - 0.475i)T \)
43 \( 1 + (0.371 - 0.928i)T \)
47 \( 1 + (-0.986 - 0.164i)T \)
53 \( 1 + (-0.909 - 0.416i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.980 + 0.197i)T \)
67 \( 1 + (-0.909 - 0.416i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.601 - 0.799i)T \)
79 \( 1 + (0.652 + 0.757i)T \)
83 \( 1 + (0.894 - 0.446i)T \)
89 \( 1 + (0.894 - 0.446i)T \)
97 \( 1 + (0.922 + 0.386i)T \)
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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

−23.73527378136236316338223831498, −23.13157857009701658111054980473, −21.89389264212111745481162244298, −20.50205270622794118296096824653, −19.72672335030068472463318254954, −19.23926743431808310260602198625, −18.258530414169507206139802927263, −17.44742063144455784403475344184, −16.735823912332113602256791128214, −16.285045872741384044185188238665, −14.77377967841164812840655470427, −14.34498933921416688912044121803, −13.31874698223611271321249456624, −12.108710731861236262867809862852, −11.50953873791068950312758222075, −10.56075661201022129187409332048, −9.20965170689501396400591170157, −8.25658618761778856496993495347, −7.89309385499946117026143403612, −6.80589499639586422107328689421, −6.10714212597872234373316912773, −4.84814402020182541107057167567, −4.02416179496627555698314459145, −1.68506698003999624469050643611, −1.31541422291803434411159558541, 0.45102562435748156138736812885, 2.26078706328676637050837926114, 3.24782709105276965866326984047, 4.11890386623681193575249667019, 5.04575667341941314015557301010, 6.48456905363500285560793498606, 7.66543695719785414786970119375, 8.53820357056139933815293029301, 9.32823822039835576566083885555, 10.335101284071731882213374296877, 11.122390971988474039078611297352, 11.62920424049143253323780586394, 12.33497242826997100430212934603, 13.9224274946127567574925542510, 14.824341165209537270339796039292, 15.63229664265983064779673003854, 16.36970209145687502783238032998, 17.65438249217946780977964068626, 17.83868948911849303957706640796, 18.99783681108063056755028193987, 19.98881022487748868288841325492, 20.388392910938646046785221990, 21.530014574571433224273080952047, 22.1414062401208154355017195680, 22.62846585132407848005697672654

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