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Properties

Label 1-571-571.146-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: $\chi_{571} (146, \cdot )$
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

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

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