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Properties

Label 1-571-571.328-r0-0-0
Degree $1$
Conductor $571$
Sign $0.411 - 0.911i$
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  + (−0.191 + 0.981i)2-s + (−0.998 − 0.0550i)3-s + (−0.926 − 0.376i)4-s + (−0.298 − 0.954i)5-s + (0.245 − 0.969i)6-s + (−0.986 − 0.164i)7-s + (0.546 − 0.837i)8-s + (0.993 + 0.110i)9-s + (0.993 − 0.110i)10-s + (−0.592 + 0.805i)11-s + (0.904 + 0.426i)12-s + (0.851 + 0.523i)13-s + (0.350 − 0.936i)14-s + (0.245 + 0.969i)15-s + (0.716 + 0.697i)16-s + (0.716 + 0.697i)17-s + ⋯
L(s)  = 1  + (−0.191 + 0.981i)2-s + (−0.998 − 0.0550i)3-s + (−0.926 − 0.376i)4-s + (−0.298 − 0.954i)5-s + (0.245 − 0.969i)6-s + (−0.986 − 0.164i)7-s + (0.546 − 0.837i)8-s + (0.993 + 0.110i)9-s + (0.993 − 0.110i)10-s + (−0.592 + 0.805i)11-s + (0.904 + 0.426i)12-s + (0.851 + 0.523i)13-s + (0.350 − 0.936i)14-s + (0.245 + 0.969i)15-s + (0.716 + 0.697i)16-s + (0.716 + 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2766692280 - 0.1785997697i\)
\(L(\frac12)\) \(\approx\) \(0.2766692280 - 0.1785997697i\)
\(L(1)\) \(\approx\) \(0.4910009097 + 0.1059604905i\)
\(L(1)\) \(\approx\) \(0.4910009097 + 0.1059604905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.191 + 0.981i)T \)
3 \( 1 + (-0.998 - 0.0550i)T \)
5 \( 1 + (-0.298 - 0.954i)T \)
7 \( 1 + (-0.986 - 0.164i)T \)
11 \( 1 + (-0.592 + 0.805i)T \)
13 \( 1 + (0.851 + 0.523i)T \)
17 \( 1 + (0.716 + 0.697i)T \)
19 \( 1 + (-0.998 - 0.0550i)T \)
23 \( 1 + (0.546 + 0.837i)T \)
29 \( 1 + (0.0275 - 0.999i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (0.851 - 0.523i)T \)
41 \( 1 + (-0.298 - 0.954i)T \)
43 \( 1 + (0.993 - 0.110i)T \)
47 \( 1 + (-0.962 + 0.272i)T \)
53 \( 1 + (-0.191 - 0.981i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (-0.754 - 0.656i)T \)
67 \( 1 + (-0.754 + 0.656i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.0275 + 0.999i)T \)
79 \( 1 + (0.137 - 0.990i)T \)
83 \( 1 + (-0.962 + 0.272i)T \)
89 \( 1 + (0.716 + 0.697i)T \)
97 \( 1 + (-0.926 - 0.376i)T \)
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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

−23.03731434114527521584840102625, −22.65857473984907395147702972887, −21.68516365985738579243216098911, −21.200469884813723016486810218086, −19.97215816220514284472853543563, −18.933218020889458134787075855942, −18.56487844426835008211698937147, −17.93675837333799935950637173113, −16.66439344718912516163391727760, −16.11083284033653099206688434580, −15.00979830557262540539305324274, −13.76345450694881661422505807550, −12.91116792425535597431698731581, −12.2251929779668612456713455357, −11.181542774545332007185296837402, −10.64442396961788327567005409999, −10.06619327680244736744654249807, −8.866728358412229492558353677739, −7.73655463594139720646167935070, −6.55653626028960343336234337478, −5.77051224267499224865313650875, −4.56481574774602587873171942617, −3.332882837786472730452936479448, −2.824981522578618285912306407373, −1.04483324782644461199562489541, 0.267068345396507431320692634937, 1.56017224922866033642435909920, 3.894043542066804615223474137680, 4.51060839798068155337173400546, 5.656264250106723206616143707118, 6.22044915865812438886927053671, 7.27250763044640020854355620357, 8.09085113520198982440787403452, 9.29450478304397903368363848332, 9.92861167490196868065173240801, 10.97953880233520827186154138646, 12.2925253129905023227643564281, 12.953623330757192697582764610109, 13.49162630185497999829479420885, 15.12681771769252917353463655252, 15.72321379900239537596373113502, 16.45922781948930302070285598586, 17.04561160354281759142754765609, 17.72900026444315332384627147082, 18.94141962177208479204303760627, 19.30245001995314363354305491312, 20.78229657678166482763507981523, 21.56653322456535340131090701431, 22.7552566454427206360827912308, 23.32558453842161047373172372997

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